The France international has emerged as one of a number of targets up front for the Hammers, who have been short of options in that area due to persistent injuries suffered by Andy Carroll.
Giroud has largely impressed during his time at Arsenal, though not quite enough to be considered a regular starter, particularly as he increasingly lost his place in the first XI last season.
Arsenal now look to be closing in on the signing of Lyon’s Alexandre Lacazette to strengthen up front, which could mean the end of Giroud’s Emirates Stadium career.
The 30-year-old has stressed he is eager to play more often next season ahead of next summer’s World Cup with France, and Winterburn believes West Ham could be the ideal move for both parties.
‘I think Arsenal have stated that they would like to keep Giroud, but they might need to review the situation depending what happens in the summer,’ Winterburn told beIN Sports.
‘He’s probably looking at his age and thinking “I need to be playing and I want to make sure that I’m part of the French squad”.
What Giroud and Wenger have been saying
Giroud: ‘I have had limited playing time for several reasons but that is not to say that I lost the confidence of the coach. I will not be content with limited playing time next season.’
Wenger: ‘It has been a frustrating season for him because he has not played a lot. And when he has played well, he has not always played the next match. He has won us a lot of points when he has come on because he has scored lots of times. He has qualities that our team need.’
‘I think he’s done a very good job and I just wonder whether Arsenal want a little bit more mobility going forward. He found himself on the bench at the end of the season and he obviously wouldn’t want that to happen.
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‘He might also be settled in London and if that is the case, West Ham have a good link with Arsenal and if he wanted to stay in London that could be a good opportunity for West Ham to get a very good striker.
‘I could see it happening but you have to think whether the player believes he is capable of playing for a top-six club and West Ham are not a top-six club. It’ll be a case of whether West Ham can convince the player to join them.’