The Rays have always said they're open to anything. This would qualify as anything.
Jim Thome turns 40 in late-August and bats lefty while spending time with the White Sox and Dodgers. There's no need to overstate this: he's a hitter. Either as a DH or substitute bat, Thome goes as his bat does. Despite his age, he can still hit. Over the last three seasons has posted wOBA of .367, .370, and .410 while playing nearly full-time. Against righties his line is .274/.402/.551. Reread that slash line. Those need to be adjusted a bit, since he played in hitter parks and not against the American League East, but that's still pretty impressive.
As best as I can tell, there's four roster layouts the Rays can use:
A. 1 DH, 2 MIF, 12 P
B. 2 DH, 1 MIF, 12 P
C. 2 DH, 2 MIF, 11 P
D. 1 DH, 2 MIF, 11 P, X 1B/COF
(The last layout would include Dan Johnson, or whomever, as the 14th bench player.)
Take Thome versus righties and add it to Burrell versus lefties (.249/.391/.457) and you have one heck of a DH platoon. There are some concerns though.
1. No reserve shortstop
I literally just wrote about this last week. With Thome on board, the Rays would have Ben Zobrist as the back-up shortstop. An injury to Jason Bartlett means Reid Brignac is on the first thing going 60+MPH towards St. Petersburg anyways, but that would prove to be inconvenient on scheduled off-days. This would also make Willy Aybar the reserve first, second, and third baseman.
2. Burrell's salary
You can't really worry about this anymore. If he's on the Rays roster and not traded, then he's getting the money. It's a sunk cost. He's probably going to bounce back, but Thome + Burrell adds more value than Burrell + a back-up shortstop. The 13th positional player on this roster just isn't getting that much time, no matter who it is.
3. Thome's salary
Here's where things get tricky. One has to figure Thome wants a bit more than league minimum. If he were to provide 90% of his .953 OPS through, say, 450 plate appearances, he would be worth nearly two wins offensively. Without breaking out a spreadsheet or calculator, let's just say he probably adds wins 95 and 96, which is insanely valuable in this division. That allows the Rays to pay more than market value for those wins, but frankly I'm not sure they'd have to since nobody else is willing to pay a lot for DH types.
4. Dugout jokes
These are worth something, right?
MANNY ASKED OL JIMMERS IF I WAS GONNA GROW CORNROWS IN HOLLYWOOD BUT I HAVEN'T WORKED ON THE FARM SINCE I WAS IN HIGH SCHOOL
WHY IS J.P. HOWELS WARMING UP ON THE ISLAND FROM THE LOSTS?
I LIKE TRIPS TO TAMPA A LOT BETTER SINCE THE DEVIL LEFT TOWN *CHORTLE*
With all the said, having two DH on one roster seems so wasteful and inefficient. But again, the price might be too good to pass up. There are other questions to tackle too. How will Thome's bat translate to the ALE? Is there any reason to believe his lackluster second half is a sign of things to come? Does he still have good bat speed? And so on. If the Rays have some interest, I would trust they've done due diligence.