The 24-year-old striker, Rovers’ joint top scorer this season with five goals, had to be substituted during the second half of Saturday’s 1-0 defeat at Charlton Athletic.
May had started the last five league games but won’t now return until December at the earliest.
“He’s having an operation on Wednesday and he’ll be out for a minimum of eight weeks,” said boss Darren Ferguson.
“It’s a hernia problem, so that’s a blow. He’s been carrying it for a little bit to be fair.
“It’s really disappointing for him because he’s having a decent season. He’s a real fans’ favourite and he’s had a really good run of games in the team.
“But we couldn’t let it go on any longer. On Saturday he couldn’t sprint and when that happens he just looked over to the bench and said ‘I can’t even sprint now.’
“So we’ve got him in quick but it will be a minimum of eight weeks.”
He added: “We’ve managed it as best we can.
“We hoped to manage it in a way that he didn’t have to have the operation.
“But given the fact he’s played a lot of games, and there’s been so many games over such a short period of time, it’s just caught up with him really so he’ll have to get it sorted.”
Liam Mandeville is Rovers’ only fit striker to face Portsmouth.
John Marquis serves a one-game suspension, while Andy Williams remains sidelined with a thigh injury.
James Coppinger, who went off early against Charlton with a calf problem, will have a fitness test during the build-up to the game but is rated very doubtful.