Hogg has spent his entire professional career at Glasgow but his contract expires at the end of the season and Rennie expects him to be a target for wealthy French and English clubs.
"We're keen to keep him here long term and those discussions are going on," Rennie told BBC Scotland.
"He's probably the highest-profile player in Scotland. I'd imagine there will be a number of big clubs who will target someone of his ability.
"I think he's enjoying it here and he wants to stay. Certainly we in Scotland want to keep him here, so we'll see how that pans out over the next few months."
Rennie does not accept that Hogg, who has two young children, will inevitably follow the path of Scotland team-mate Finn Russell, who joined Racing 92 this summer.
"It's always different for different people," said the New Zealander. "I always felt it was a good move for Finn and he was keen to go. He was keen for a change, so that was always going to be a done deal.
"The situation with Hoggy is probably a little bit different.
"It's very difficult to compete with the French clubs who are offering two, three times what we would offer players," he added.
"So we've got to focus on environment, culture, and hope players will enjoy playing the type of footy we play and a team with good men where their game can develop and maybe cash in at the end of your career.
"We want to keep people like Hoggy here but you know we've got budgets and salary caps that we've got to work to and it's well under what's happening in France and England.
"I guess we've got to prioritise - if you pay a lot for one player then you need to cut back on other areas. The only other option is to spend more money on players or consistently try and bring through good young kids through the system which costs you less."