Nothing's official yet, but with Damon himself confirming the deal, let's assume that it's going to happen.
Damon has even answered playing time questions already:
Damon noted that the Indians have indeed offered him a role in left field.
"That's what Cleveland has offered me," he said. "I need to go out there and show people I can still play there. Right now, Shelley Duncan is the starting left fielder. I know they are going to rotate me in and give him a breather but, you know, I also understand this game, too. If Shelley Duncan is on fire at the plate, he's going to play. And if I am, they are going to find a place for me.
However, one of the interesting aspects to the deal that's almost signed is a combination of a no-trade clause and an out-clause that would be tied to Sizemore being activated from the Disabled List. I can't remember a major-league contract ever containing a clause like this, especially one that's tied to another player's health. So if both these clauses are included, the Indians not only cannot keep him around as a fourth outfielder, they can't get anything for him if they would wish to trade him.
Adding Damon's bat is a good move. As Tampa's DH last year, he hit .261/.326/.418, which was good for a 110 OPS+. But with the Indians, Damon will be a left fielder, and a defensive liability. Teams will run at will on his arm, but that's always been the case. Another concern is whether he can actually play the position on an everyday basis; he hasn't logged more than 40 games in the outfield in a season since 2009.
If the Indians do sign Damon, then that probably means Aaron Cunningham is gone (he's out of options), though that would mean the Indians would be without a true backup in center field. They could also opt to keep Cunningham and go with six relievers for the time being.
UPDATE: Jon Heyman says that the deal would be $1.25M plus up to $1.4M in bonuses, and also that the contract is complete.