The 33-year-old was shown round the club's Auchenhowie training ground on Saturday. Talks will continue and a deal could be imminent.
Barton was named in the Championship team of the season as Burnley secured the title.
He has yet to agree a new deal with Burnley and could be tempted by a longer term contract at Ibrox.
Rangers manager Mark Warburton is keen to add some experience and character to his squad in preparation for their return to the Premiership.
Barton, who won one England cap, was named player of the year at Burnley this season.
The former Manchester City, Newcastle, QPR and Marseille player joined the club when Frank McParland was the sporting director.
McParland is now head of recruitment at Ibrox, and will be an influential figure as Barton weighs up his options.
The transfer runs counter to Rangers' recruitment strategy of signing players with the potential to develop and rise in value.
But Barton is a player McParland knows well and Warburton prefers to sign individuals whose nature and character he is already aware of.
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Rangers have made a move to sign Burnley midfielder Joey Barton on a free transfer this summer.