Jayasuriya, 55, said he had been asked by Sri Lanka Cricket to coach the national squad ahead of a white ball series against India this month, and Sri Lanka's tour of England in August.
"I have been asked to take over coaching and I am happy to do it," Jayasuriya told AFP.
Jayasuriya, a former Test and ODI captain, had also been a national selector.
He is to take over the team after the end of the ongoing T20 Lanka Premier League on July 21.
Jayasuriya represented Sri Lanka 586 times across the three formats of the game, hitting 42 centuries and taking 440 wickets. He played his last international in 2011.
Former England coach Silverwood, 49, quit for "personal reasons", according to Sri Lanka Cricket, which thanked him for his "valuable contributions during his tenure".
Sri Lanka won only one match at the T20 World Cup tournament, against the Netherlands, and finished third behind South Africa and Bangladesh in their group.
Silverwood took over in April 2022 and enjoyed immediate success, as Sri Lanka won the T20 Asia Cup later that year.
They reached the final of the 50-over Asia Cup in 2023.
But they failed to make it to the knockout stages of the T20 World Cup in 2022 and last year's 50-over World Cup.
Former skipper Mahela Jayawardena also stepped down as "consultant coach" last month.
