Parker is a doubt for the competition in Poland and Ukraine this summer and the England stand-in captain's absence would be a blow for new coach Roy Hodgson.
However, Spurs manager Harry Redknapp confirmed after their 2-0 victory over Fulham that the midfielder should make the tournament should the injection go according to plan.
He said: "Parker has an Achilles injury, he's going to have an injection tomorrow and they're hoping with about six or seven days' rest it could clear up."
However, Redknapp warned that if things did not go to plan then Parker would require surgery - ruling him out of action.
"We'll have to wait and see," he added. "If it doesn't clear up then 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."