Warner scored 193 for Australia A against South Africa A in Pretoria last week after being sent to join the second-string side's tour to gain match practice.
The 26-year-old was suspended by Cricket Australia last month after punching Joe Root in a Birmingham bar following a Champions Trophy match at Edgbaston.
He was then left out for the opening two Test defeats at Trent Bridge and Lord's but could now come back into consideration having rejoined the squad.
And Lee believes Warner is the right man to spark some life back into Australia's tour.
He said: "I'd like to think they can get a win in the third Test coming up and maybe if they inject someone like a David Warner - a pure match-winner - I reckon that'd be a great way to get the Australians on the board."
Lee added of Australia's chances of an upset victory in Manchester: "I would never back against Australia but I think we've got a lot of hard work ahead of the guys.
"Australia have been playing some decent cricket but we've just fallen short of the mark and I think the way that England have been playing, they're playing purely on confidence now.
"Graeme Swann is bowling with pure confidence, and they've got the batsmen who are backing that up.
"We probably haven't had that with the Australian team right now so it only takes one huge innings or one good bowling spell to get the boys back in the right frame of mind."
Lee believes Australia will perform better on home soil when England go Down Under this winter, adding: "I said before the tour started I reckon that England will be very strong in their home country but when we come out here with the Kookaburra ball that the guys are used to playing with and playing on the wickets here that are more conducive to the fast bowlers, I reckon we'll be a lot stronger."
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- David Warner
- Cricket Australia