After holding talks with Lescott and agreeing a £1.5million fee with Preston for striker Joe Garner, Warburton confirmed he was “hopefully very near a conclusion on one or two targets”.
The Rangers boss added at his Thursday lunchtime media conference: “I hope we can get them to completion in the next 24 to 48 hours max.”
Warburton met Lescott in Manchester on Tuesday in a bid to persuade him to move to the Ladbrokes Premiership after the 34-year-old former England centre-back was told he was free to leave Aston Villa.
When asked about his chances of signing a player of Lescott’s calibre, Warburton said: “I’m optimistic because we have to keep raising the profile of Rangers.
“Players know about the club, know about the history and stature of the club and hopefully we can keep pushing forward. If we can do that and sign players of the right calibre, then it’s almost self-fulfilling.”
Sunderland are also reported to be interested in Lescott but Warburton believes Rangers can bridge the financial gap between the Scottish and English top flights by stressing the experience on offer at Ibrox.
“We can’t compete with Premier League clubs, absolutely not,” he said.
“The financial disparity is evident for all to see. But what we can do is look at what we can offer a player. We can offer a player an opportunity here that is maybe not on offer at that club. And maybe those clubs can sign more players to have bigger squads.
“If a player doesn’t buy into that, no problem, we shake hands and move on. You want players who want to be here, recognise an opportunity and see a role for themselves with Rangers.
“We can’t compete when it comes down to straight cash, but with players of experience and players of quality it’s not always about a straight cash deal. It’s very much about the quality of opportunity. If we can get that right, then we have a chance.”
Garner looks certain to join after telling Preston he wanted to leave. The 28-year-old netted six goals for the Sky Bet Championship side last season after hitting 27 in their promotion campaign the previous term.
Warburton said: “A player like that adds experience, adds an edge in terms of how he plays the game, in terms of his enjoyment of the game, a variety of goals, comfortable inside and outside the box, movement and a threat. He’s a constant threat.”