The midfielder, who captained England in their last game against the Netherlands in February, has an Achilles injury which his Tottenham boss has described as "a long-term concern".
The Spurs manager says the only cure could be rest but Parker is likely to be wanted by new England boss Roy Hodgson for next month's tournament in Poland and Ukraine.
Parker has been left out of the Tottenham starting XI for their last three matches and it is unlikely he will be selected for Sunday's season finale against Fulham - a game the Londoners need to win to secure at least a top-four finish in the Barclays Premier League.
Redknapp thinks the 31-year-old may have to miss the Three Lions' pre-Euro 2012 training camp, which begins on May 21 in Spain, and fears for his fitness over the summer.
He explained: "He's had a problem with his Achilles, I hope it's not going to get worse. It isn't getting any better. It's a long-term concern.
"The way he is at the moment, he is struggling for fitness.
"It's not a rupture but it is sore. It is bothering him.
"After tomorrow he will have some injections in it and we'll see if that can clear it up.
"He's been carrying it a few weeks. Maybe the only cure is a bit of rest but maybe the injections might clear it up.
"The specialist thinks it will clear up but it will need 10 days' rest afterwards."