Barton is on the books of Premier League-bound Burnley at the moment, but his current contract with the Clarets runs out in June 2016.
According to BBC Sport, Rangers have made a move to sign the former Newcastle United star on a free in the summer transfer window.
The Gers, who will play in the Scottish Premiership next season after winning the Scottish Championship title this campaign, showed Barton around the Auchenhowie training ground on Saturday, according to the report.
Former Rangers attacker Lovenkrands has said that Barton would be a good signing for Mark Warburton’s side.
The 36-year-old former Denmark international, who played with the Englishman at Newcastle, believes that the 33-year-old would fit into the Gers’ playing style.
“Joey has played on the big stage and he’s a hard worker - on and off the pitch,” Lovenkrands told The Scottish Sun. “If he signs for Rangers, it will be interesting to see what kind of role he fills and how he fits into the style.
“Joey is better in a central area. He has played out wide but the formation Rangers play and the Scottish league should suit him.
“Joey would be a really good signing for Rangers and it would make next season even more interesting.”
Signing Barton would be a blockbuster move for Rangers, who have won the Scottish Championship title this season and will play in the Scottish Premiership during the 2016-17 campaign.