Erdogan told Abbas Ankara reached a deal with Tel Aviv on improving humanitarian conditions in Gaza, according to Turkish presidential sources who spoke on condition of anonymity due to restrictions on talking to media.
Abbas said his government welcomed the developments, the sources added.
Turkish Prime Minister Binali Yildirim is due to make a statement Monday at 1 p.m. local time (1000 GMT) regarding progress in the negotiations with Israel, sources at the Prime Ministry told Anadolu Agency.
In 2010, six civilian ships in a humanitarian aid flotilla were attacked in international waters by Israeli forces.
The vessels had been trying to break Israel's blockade of the Gaza Strip.
Nine Turkish citizens were killed and 30 other people injured; one person died nearly four years later after being critically injured in the attack.
Turkey had demanded an official apology from Israel, compensation for the families of those killed in the attack as well as the removal of Israel’s blockade on Gaza in the aftermath of the attack.
In 2013, Israeli Prime Minister Benjamin Netanyahu apologized to then Turkish premier and now President Recep Tayyip Erdogan for the attack.
In recent months, the two countries have been engaged in talks aiming at normalizing ties.