The proposed two-month extension will require parliamentary approval.
The European football competition involves 24 national teams and runs from 10 June to 10 July.
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The coordinated gun and bomb attacks in Paris on November 13 killed 130 people and were claimed by so-called Islamic State.
The Stade de France football stadium, which was targeted by suicide bombers, will host the opening match of Euro 2016 and the final.
The current state of emergency, which gives police extra powers to conduct searches and put people under house arrest, runs until 26 May.
It was last extended for three months in February, despite concerns raised by rights groups that which said police had abused their powers.
Interior Minister Bernard Cazeneuve will seek other ministers` approval of the extension later on Wednesday, before asking parliament to vote on it, Mr Valls said.
"The state of emergency cannot be permanent, but on the occasion of these big events… we have to prolong it," he said.
The proposed extension would also cover the Tour de France bicycle race.
Security will be tight at the Euro 2016 tournament, with more than 1,200 security officials deployed for matches at the Stade De France, and an average of 900 across all stadiums.
Mr Cazeneuve said on Friday that more than 3,500 searches and 400 arrests had been carried out since the state of emergency was first imposed in November.
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