Arsenal this week completed a club-record deal for striker Alexandre Lacazette from Lyon, raising doubts over Giroud's future at the Emirates.
Giroud struggled for game time at the Gunners last season, starting just 11 Premier League matches.
He has been linked with a return to France with Marseille and Lyon, while Premier League rivals Everton and West Ham are said to be interested.
And bookmakers Betfair have slashed Giroud's odds of joining Marseille this summer from 12/1 to 5/1 following a flurry of bets.
Giroud admitted last month that he could be tempted into joining Marseille.
"I am not insensitive to the approaches of the French clubs," he told broadcaster Telefoot. "Especially that of a legendary club like Marseille."
But Everton remain 5/4 favourites to sign the France international as they look to replace last season's top goalscorer Romelu Lukaku.
Manchester United announced yesterday that they had agreed a fee with Everton for Lukaku, believed to be £75million rising to £90m.
Giroud joined Arsenal in the summer of 2012 from Montpellier and has gone on to win three FA Cups, scoring 98 goals in the process.
And Betfair have priced the the Frenchman at 7/4 to remain at the north London club for another season, while West Ham are at 5s.
Source: Daily Express