Sri Lankan batsman Kusal Mendis hit a defiant 77 to block England's path to a landmark win in the third and final Test on Monday.
Chasing a daunting 327 for victory to prevent England scoring a series whitewash, the hosts were 164 for five at lunch on day four. Mendis and Roshen Silva, on 37, were batting.
Sri Lanka need another 163 runs to avoid a humiliating third straight defeat in the series. England are looking for their first overseas sweep in a series of three or more Tests since Ted Dexter's England won 3-0 in New Zealand in 1963.
Left-arm spinner Jack Leach got the only wicket of the morning session after Sri Lanka resumed on 53-4. Nightwatchman Lakshan Sandakan scored seven before giving Ben Stokes a catch.
Mendis, who started the day on 15, has put on 82 runs for the sixth wicket with Silva.
Moeen Ali had led England's spin charge in the final session on Sunday, removing the Sri Lankan openers for just 24 runs.
Leach and Stokes also took a wicket each on Sunday as Sri Lanka slipped to 82-4. England were bowled for 230 in their second innings with Jos Buttler making a crucial 64 after they had fallen to 39-4 against the opposition spinners.