The Indians enter the series - and a six-game road trip - in second place in the division, trailing first-place Detroit by two games and leading third- place Chicago by 1 1/2 games.
Brought over by trade from the Colorado Rockies at the non-waiver trade deadline, Jimenez has started twice for the Indians and has gone 1-0 with a 3.48 earned run average while allowing 12 hits in 13 innings against Texas and Detroit.
Jimenez, a 19-game winner with the Rockies last season, was 6-9 with them in 21 starts before the trade, including a no-decision against the White Sox after a seven-inning outing in Denver on June 29.
He gave up two runs on seven hits in Colorado's 3-2 loss - his lone career matchup with the White Sox.
Jimenez will be faced by 28-year-old righty Gavin Floyd, who's won four of his last five starts since beginning the season 6-9.
The former first-round draft choice of the Philadelphia Phillies (fourth overall, 2001) came back after the All-Star break with an 8-2 win at Detroit and subsequently beat Cleveland and Boston before a loss to the New York Yankees on Aug. 3.
He rebounded with win No. 10 of the season, a 4-3 decision at Baltimore, where he allowed three runs on seven hits in 6 2/3 innings.
Lifetime against Cleveland, Floyd is 6-3 in 10 starts with a 4.25 earned run average in 59 1/3 innings. In the July 22 game, he tossed 7 2/3 scoreless innings and allowed four hits in a 3-0 triumph.
Danks (5-9) took a no-hitter into the sixth inning before losing the bid on a single by Billy Butler. He gave up two runs on four hits while fanning six and issuing four walks. Will Ohman worked a scoreless ninth to seal the win for Chicago.
The Indians, who had the finale of their three-game set with the Minnesota Twins postponed by rain on Sunday, have won four of their last five games.
Chicago has won six of its seven meetings with the Tribe this season.