"He has got an Achilles injury and will have an injection tomorrow and they are hoping with six to seven days rest it will clear up," Spurs manager Harry Redknapp said on Sunday after his side finished fourth in the Premier League following a 2-0 win over Fulham at White Hart Lane.
"If it doesn't clear up he wouldn't be fit (for the Euros). If he had to have an op then it's going to be long term. We'll have to wait and see how he is."
Parker, who wore the captain's armband in England's last game against the Netherlands in February, had been enjoying a superb season before the injury and established himself as first choice for England in the holding midfield role under former manager Fabio Capello.
New England manager Roy Hodgson may have to work without the 31-year-old at England's pre-tournament training camp, which starts on May 21 in Spain.