Kılıçdaroğlu had asked Bahçeli for the meeting, the source said.
The first round of voting over the constitutional reforms has been completed with the parliament passing a total of 18 articles — each with over 340 Yes votes.
On Wednesday, lawmakers will go to the polls to vote on each article for a second time and on the entire package.
If the package passes the parliament with votes of at least 330 of 550 deputies, the new articles will be put to vote in a national referendum.
If the reforms obtain the support of 367 lawmakers, it can pass into law without a referendum although President Recep Tayyip Erdogan said he would push for a referendum even if the draft is approved by the two-thirds majority.
Proposed changes to the constitution require a simple majority (51 percent).
The ruling Justice and Development (AK) Party has 316 seats and the MHP — which has so far supported the constitutional change — has 39 seats.
The Yes votes from the two parties would be enough to secure a referendum, which is likely to be held at the beginning of April.