Jayawardene picked up the injury while fielding in the slips during the Singhalese Sports Club's Premier league match against Nondescript's Cricket Club.
"Initial scans showed that it was only a dislocation but the scans were sent to Australia for further opinion," Sri Lanka team manager Michael de Zoysa said.
"The opinion from Australia is that Mahela will have to undergo surgery and have a wire inserted to keep the dislocated bone in place.
"He will be out for between four-to-six weeks which means he is certainly ruled out of the Bangladesh Test series."
Thilan Samaraweera would replace Jayawardene for the two Tests against Bangladesh, Sri Lanka Cricket said in a statement.
Another former captain Kumar Sangakkara is also fighting to recover in time for the first Test starting in Galle on March 8.
Sangakkara had his left index finger broken by Australian fast bowler Mitchell Johnson during the Boxing Day Test in Melbourne and had a pin and plate inserted for his injury.
The left-handed batsman has begun having light batting sessions in the nets but is still uncertain for the Galle Test.
Jayawardene and Sangakkara are Sri Lanka's two most prolific run-getters in Tests and one-day internationals, having scored over 10,000 runs each in both formats.
Bangladesh will also play three one-day internationals and a Twenty20 international on the tour.