The second ship is reported to have more than 100 people on board.
Sri Lankan President Anura Kumara Dissanayake has convened advisers to consider a request from Iran to allow the warship to enter Sri Lankan territorial waters. Iran fears the ship will otherwise suffer the same fate as the IRIS Dena, according to sources.
On Wednesday, a US submarine launched a torpedo attack on the IRIS Dena, which had a crew of 180.
About 30 Iranians were rescued from the sinking ship and 87 bodies have been recovered so far.
Iranian Foreign Minister Abbas Araghchi has written on social media that the US will "bitterly regret" the torpedo attack.
