The free agent was pictured being shown round the Glasgow giants’ stadium and Murray Park training base last Saturday.
The 33-year-old - named Sky Bet Championship winners Burnley’s player of the year - has been offered a 12-month contract extension by Clarets boss Sean Dyche but is also mulling over Warburton’s proposal of a longer-term deal with Gers.
The Ladbrokes Championship winners believe the one-time England midfielder will give them the added steel they require ahead of returning to the top flight next season.
But Warburton confessed he has yet to be given an indication of Barton’s thinking.
Asked if he was still waiting for a response, the Gers boss said: “Yeah basically.”
However, Warburton - speaking at the pre-match media conference for Saturday’s William Hill Scottish Cup final with Hibernian - remains confident.
He added: “Joey is a player that will have many, many suitors I’m sure. Our job is to sell the opportunity that Rangers represents.
“We’re never going to compete with wages down south, that’s an obvious point to highlight. But if we can show what we are as a club, the support, the history, the stature of the club and what our aspirations and targets are then hopefully any player will find that appealing.
“If they do hopefully we can conclude business. If not, we move on.
“We were delighted to welcome Joey to Glasgow last weekend. He was great company. He’s a very experienced, a very combative and highly intelligent footballer, so we were delighted to talk to him.
“I’m sure he will have many suitors but our hope is that we can get it over the line.”
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