Jimenez recently told the Cleveland Plain Dealer that he hasn't formally opened contract talks with the Indians but "would listen" to their offer.
"We haven't talked, but of course I would listen to them," Jimenez told the paper. "They have treated me really good here. It feels like home."
Kazmir also indicated that he has enjoyed playing in Cleveland and would like to return to the Indians.
"I would love to stay here," Kazmir told the Plain Dealer. "I'm very interested ... I'd love to contribute and continue to be with this group."
Jimenez, 29, has enjoyed a bounce-back campaign this season, his second full year in Cleveland. The hard-throwing right-hander has an $8 million mutual option for 2014 with a $1 million buyout.
Following a disappointing 2012 season when he led the majors with 17 losses and posted a career-worst 5.40 ERA, Jimenez is 9-9 with a 3.95 ERA and 143 strikeouts in 141 1/3 innings this year.
Kazmir, 29, is 7-7 with a 4.36 ERA this season, his first full year in the majors since 2010. The left-hander signed a one-year, $1 million contract with the Indians this past offseason.
The Indians (72-65) are on pace for their first winning season since 2007 and entered Tuesday just 3½ games out of the American League's second wild card.