The midfielder injured his right shoulder in stoppage time of Leeds's final game of the season, a 3-1 win over West Bromwich Albion. He had earlier scored his first Premier League goal from a free-kick.
Phillips did not appear for the post-match lap of honour and the injury was of particular concern because he dislocated his shoulder in the autumn.
England manager Gareth Southgate is due to name his 26-man squad for the tournament on Tuesday, although Uefa's deadline is June 1.
Since making his debut in September's friendly against Denmark, Phillips has won six more caps, all but one from the start.
Whether he would have started in England's competitive midfield or not it seems inconceivable he would not be picked if fit, so Leeds's medical update will be well received in Harrogate, where Southgate lives.
It said that the injury "is currently being treated conservatively (and) we remain confident he will be available for Euro 2020, if selected for the England squad."
England's opening game is against Croatia on June 13. They have friendlies against Austria and Romania at Middlesbrough on June 2 and 6 respectively.
Phillips's club-mate Tyler Roberts has been selected for a Wales training camp, but there have been differing fortunates for the Spanish pair of Diego Llorente and Rodrigo.
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