If a costly Scherzer fetched a roughly #50-ranked prospect (#44 by MLB.com, #56 per Fangraphs) and Giolito did the same (#65 per MLB; #61 per Fangraphs), I'm certainly curious what a dirt-cheap Paxton would bring back, even if we have to replace him.
But the number of trade possibilities are finite. I assume any Paxton deal would be limited to NL teams, and I assume that the main return wouldn't be a catching prospect. With all that in mind, and using Scherzer/Giolito returns as a baseline, here are 10 NL teams and their trade targets, listed in descending order of playoff odds...
Locks
The Braves have a terrible system and are getting Fried and maybe Wright back soon. I think the only deals that seem to work here involve AJ Smith-Shawver (#67) or Vaughn Grissom, and I doubt they do it.
The Dodgers already traded for Lynn, but have to be pretty worried about Gonsolin and Urias. They have multiple prospects in the mid-Top 100 range (Michael Busch, Andy Pages, Gavin Stone), and post-prospect Miguel Vargas seems like a candidate to be moved too.
Likelies
The Phillies have the best pitching staff in baseball, so wouldn't seem to need Paxton. But they also need an outfielder. Would they give up Mick Abel for both? Doubtful.
The Brewers are getting Woodruff and Miley back, so may be less inclined to trade for Paxton. But they too need an outfielder, and Verdugo would slot into a nice platoon situation with Wiemer. The guy I like there is Jacob Misiorowski (#84 on MLB's Top 100), and 2B/CF Tyler Black has immense plate discipline and speed.
The Giants aren't giving up Matos or Luciano, and I doubt they make Kyle Harrison available despite his rough year. I don't know how else we match up.
Hopefuls
The Marlins already have one of the better pitching staffs in the game and seem likelier to trade for hitting. On the other hand, Paxton's salary is probably interesting to them, and they have more long-term pitching assets than they can handle. Assuming they don't want to move Jake Eder (Fangraphs #54), maybe a Paxton for Trevor Rogers deal (FA 2027, currently out with a lat tear) works? Would they trade Jazz?
The Diamondbacks probably don't want to give up on Brandon Pfaadt or Druw Jones despite their 2023 setbacks. I don't think Alec Thomas or Ryne Nelson would be interesting to us.
The Cubs have a bunch of prospects in the top 100 (CF Pete Crow-Armstrong, 2B James Triantos, LHSPs Ben Brown and Jordan Wicks, OF Kevin Alcantara) but aren't likely to bet this hard on an outside bid for the playoffs. One guy who might fit here is Alcantara, a 6'6" outfielder (and former Yankee prospect, via the Rizzo trade), who is in A+ and already on their 40-man.
The Reds are a nice story and I'd love to see them go for it, but their odds are around 30 percent. Noelvi Marte and Cam Collier would be the targets here, I'd think, but I don't think they do it. There's also Edwin Arroyo, who was the #52 prospect last year, but has fallen off.
The Padres should sell, but might not. Dylan Lesko (still 19, returning from TJS) would be an interesting gamble. I don't think we'd be interested in catching prospect Ethan Salas. On their roster, a Cronenworth swap might make sense for both teams, but I doubt they move him four months after extending him.
Among American League teams, I think Minnesota is probably the only one we'd deal with. I want to say they're a lock to win the Central (and therefore not our competition), but they always make it interesting and just got swept by the Royals. They've got a very good pitching staff and wouldn't seem to need too much help, but maybe they're ready to move on from Royce Lewis?
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To recap, I see only a handful of scenarios where a Paxton deal would seem to make sense, with value consistent with the Giolito/Scherzer trades. (I'm also leaving aside the question of who would replace Paxton in our rotation, if we do a complex sell-and-buy, though I think E-Rod has some merit.)
Milwaukee receives Paxton and Verdugo, Boston receives Jacob Misiorowski, Tyler Black and Luis Urias
Misiorowski is a "high ceiling/high variance" starter who has moved quickly through the Brewers system, and Black's plate discipline is terrific. The Sox also get a chance to reboot Urias into a starting 2B. This is interesting to me but I think it's too high variability.
Los Angeles Dodgers receive James Paxton, Boston receives 2B/DH Miguel Vargas
This one makes sense to me. The Dodgers desperately need pitching and have a little bit of a surplus with these bat-first prospects (Vargas and Busch). Vargas can play a bit of 2B, 3B and LF, but he's more likely a long-term DH, a solid right-handed bat with a great approach and plate discipline, and who would be a doubles machine.
Miami receives James Paxton, Boston receives Trevor Rogers
Paxton would join Alcantara, Garrett and Luzardo as a great 1-4. The Marlins have too many young pitchers, and more (Perez, Meyer, Eder, Fulton) on the way. This swaps out one who wouldn't help them this year, but could settle in nicely for the next three years for us.
Chicago Cubs receive Paxton for Kevin Alcantara
If the Cubs go for it, I could see this happening. Alcantara is pretty high variance himself and had a rough early '23, but has been a monster over the last month after some changes at the plate. The fact that he's on the 40-man and still in A+ might make him more available.
Minnesota receives Paxton for Royce Lewis
I don't think the Twins want to move on from Lewis (who is working his way back from an oblique injury), but he's positionless in a Polanco-Correa-Julien infield and persistent injuries have left questions.
All of these guys are compelling targets, though each has some question marks about their long-term value. I think that corresponds well with Paxton, who has been the game's 21st most valuable SP since his May 12th debut, but is seen as an injury risk.