Clarke has been out with a hamstring injury and reports have suggested his future as skipper is in doubt.
"He's captain of Australia, the second most important position behind the prime minister and as such, we need our captain playing well," Lehmann said.
The World Cup, co-hosted by Australia and New Zealand, starts on 14 February.
Steve Smith has flourished as stand-in skipper in 33-year-old Clarke's absence, but the coach told Australian radio station FiveAA he was "looking forward" to his regular captain's return.
"He played grade cricket on the weekend which is great," he said. "I had a good chat to him and he pulled up really well from the grade game.
"He's ahead of schedule and I think he's said that in the press as well. Fingers crossed he might be.
"I'm sick of it," added Lehmann, who also denied rumours of a rift with Clarke. "We all get on, that's the great thing. Media can say what they like.
"From our point of view we want our captain fit, which is great leading into the most important tournament for four years.
"We've got the World Cup, the West Indies tour and the Ashes [in July and August], and we want him ready to go because he leads this side very well."
Australia coach Darren Lehmann has backed Michael Clarke as captain before the World Cup, saying he is "sick" of speculation about the batsman's future.