Fans were left scratching their heads when Mumbai opted to take Bumrah out of the attack after he bowled the fourth over
England's leading wicket-taker was named in their squad for the four-day contest at Lord's - their first against Ireland - despite picking up the injury in a County Championship match earlier this month.
"We meet on Monday and at the moment I feel good," Anderson, who has 575 wickets from 148 tests, told the BBC.
"I will keep bowling and we will monitor it. If it comes to Wednesday and I'm fit to play then great, but if not I will work towards being fit for the Ashes."
The first Ashes test begins at Edgbaston on Aug. 1, with Joe Root's side looking to regain the coveted urn on home soil after losing 4-0 away in 2017-18.
They head into the five-match series on the back of their maiden 50-overs World Cup triumph last week but Anderson said they had to put celebrations on the backburner and completely focus on red-ball cricket.
"We realise the dangers that the Australian team bring to test cricket, so we're going to have to be fully focused on that," Anderson added.
"We can't afford to hang on to the World Cup win. It's our job as English cricketers to try and carry that momentum on and hopefully an Ashes win will do that."
Fans were left scratching their heads when Mumbai opted to take Bumrah out of the attack after he bowled the fourth over
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