These AL Central foes meet in Cleveland for the first time this season Monday night.
With eight strong innings from Corey Kluber, who helped the Indians (34-34) end an eight-game skid Tuesday, Cleveland won for the fourth time in five games Sunday, 2-0 over Washington.
The victory kept the Indians one-half game in front of the Royals (33-34) for second place as Kansas City defeated Tampa Bay 5-3 later Sunday.
The Royals are 12-5 since their own 0-8 stretch.
"Kansas City is playing really well now,'' Rays manager Joe Maddon said. "... They're playing with a lot of confidence."
Kansas City has won each of James Shields' last three starts, but he hasn't picked up a victory during that stretch. Shields (2-6, 2.79 ERA) has posted a 2.14 ERA in those outings but has received three total runs of support.
He gave up two runs in seven innings of a 3-2, 10-inning victory over Detroit on Wednesday.
"Unfortunately I haven't gotten a lot of wins, but my team has won the last three games I've been out, and that's all that really matters to me," he told the Royals' official website.
The right-hander has gone 0-4 with a 2.64 ERA in eight starts since his last win April 30. His run-support average of 2.97 is among the worst in baseball.
Shields will get his first look at the Indians in a Royals uniform after not getting a turn when the teams split four games in Kansas City from April 27-29.
Cleveland will continue its nine-game homestand with Carlos Carrasco (0-2, 15.26) on the hill.
In his last start, the right-hander gave up six runs and 10 hits in four-plus innings of a 6-4 loss at Detroit on June 8.
He hasn't appeared in a game since then after serving a seven-game suspension for hitting Kevin Youkilis in his season debut, a 14-1 loss to New York Yankees on April 9. He surrendered seven runs in 3 2-3 innings before getting ejected.
Carrasco, who missed last season after undergoing Tommy John surgery, had just finished a five-game suspension after hitting Kansas City's Billy Butler on July 29, 2011, in his second to last start before getting shut down for the season.
Butler has faced Carrasco more than any other batter, going 6 for 13 with two homers. He is batting .364 with seven RBIs during the Royals' 10-2 surge.
Jason Kipnis has hit .529 with a homer, two doubles and four RBIs during the Indians' 4-1 stretch. The second baseman, who has a 10-game hitting streak and a four-game RBI streak, is 2 for 3 with a double off Shields.
Cleveland first baseman Nick Swisher is expected to miss this series after receiving a cortisone shot to help his sore left shoulder. He recently went through an 0-for-27 slump and has five hits in his last 47 at-bats.