The 25-year-old has been linked with Arsenal and Liverpool throughout the year, with both Premier League clubs sending scouts to watch him in action in Ligue 1 throughout the season.
But United are believed to be keen on bringing in a new forward this summer as Michael Owen will leave the club, and Dimitar Berbatov is also likely to leave Old Trafford, and according to Caughtoffside.com, manager Sir Alex Ferguson is ready to meet Montpellier's asking price for the in-form forward.
Earlier this year, reports in France had suggested Montpellier had raised the value of Giroud's release clause to €16million (from €10m), after learning of interest from Arsenal.
Being in a position to offer Champions League football, Montpellier are hopeful they can keep Giroud at the club next season.
But until Montpellier won their maiden Ligue 1 title last weekend, it was widely believed the France international would leave the club this summer.
Earlier this month, however, he insisted he was yet to decide on whether to seek a move this summer, or stay with the club for another year.
"I do not know yet what I'll do next season, if I leave or stay in Montpellier," Giroud told Telefoot.
"But English clubs make me dream, this championship makes me dream.
"But I am not obsessed with England! And money is not a motivation in my career.
"I make strategic choices of a sporting point of view."
Giroud is currently preparing for Euro 2012 and is the bookmakers' favourite to finish as France's top goalscorer (see France Euro 2012 Top Scorer odds) after finishing top of the pile in Ligue 1.
Who will win the race to sign Giroud this summer? Do United need another striker? Have your say below!