An £11.9 million deal for Germany striker Lukas Podolski has already been agreed and Arsenal remain confident of keeping Robin van Persie, but Wenger is still also hoping to recruit at least one other forward this summer.
Blackburn's Junior Hoilett is also on his radar but it has emerged that he has instructed Giles Grimandi, his chief scout in France, to carefully monitor Giroud for the rest of the season.
At 6ft 3in tall and around 14 stone, Giroud would certainly provide a powerful physical presence and a different sort of option to Van Persie.
The 25 year-old has been central to Montpellier’s push for the French League title this season and, with 18 goals, is the leading scorer in France.
Although clearly strong in the air, the improvement in Giroud’s all-round game has also been rewarded this season with a call-up to Laurent Blanc’s France squad.

He scored his first international goal in France’s 2-1 win over Germany at the end of February and is now expected to be a key part of the France squad at Euro 2012.
Montpellier’s valuation, however, could be a problem if Arsenal firm up their interest in Giroud with a formal offer.
In January, Newcastle were also linked with Giroud, prompting an angry response from Montpellier owner Louis Nicollin. “Newcastle? They are too small for him,” said Nicollin.
“Arsenal? You would have to see if Arsène Wenger is interested but, as before, there is Van Persie. And at €10 million [£8.3 million]? No. Given that he still has more than two years of contract, it will cost more.
“He has a contract until 2014, so €50 million[£42 million] or €60 million [£50 million] at least.”
Giroud, himself, has said that he loves the “French feel at Arsenal” but, at this stage, Wenger’s interest has not gone beyond watching the player.
In any case, the recruitment of new strikers is likely to depend on what happens this summer with Marouane Chamakh, Nicklas Bendtner, Carlos Vela and Ju-Young Park.
Vela and Bendtner are both currently on loan, while Chamakh and Park have been conspicuous this season by their collective lack of impact in support of Van Persie. Between them, they have scored just two goals.
As well as his lack of goal, Wenger is unlikely to be impressed by the emergence of a photograph of both Chamakh and QPR forward Adel Taarabt holding a shisha pipe, apparently after Saturday’s game at Loftus Road.
Shisha is a water pipe in which fruit-scented tobacco is usually burned using coal, passed through a water vessel and inhaled through a hose.
Smoking it is not illegal in England, although Wenger did previously say that it was unacceptable when William Gallas, who was then his captain, was pictured with a cigarette.
Chamakh was due back in training yesterday and, in an interview in France, has said that he still sees his future at Arsenal.

http://www.telegraph.co.uk/sport/football/teams/arsenal/9184266/Arsenal-manager-Arsene-Wenger-tracks-Montpellier-forward-Olivier-Giroud.html