The APC governors were said to have engaged in fierce argument throughout the meeting.
The meeting was said to be about the elongation of Chief John Oyegun's tenure as the party's National Working Committee (NWC) chairman.
Buhari had rejected the extension of the NWC by a year at the last National Executive Committee (NEC) meeting of the party.
Some stakeholders were said to have proposed a caretaker committee to take over from the Oyegun-led NWC and to be headed a former governor of Edo State, Adams Oshiomhole.
The party was expected to iron out the issues at the Tuesday meeting but the governors did not reach a consensus.
According to The Cable, the governors did not address state house correspondents when they left the council chamber of the presidential villa at the end of the one-hour meeting.
Governors Rochas Okorocha, Simon Lalong and Nasir el-Rufai of Imo, Plateau, and Kaduna respectively were said to have explained to journalists that a unanimous agreement was reached at the meeting not to talk to the media.
However, one of the governors, who was aggrieved, was quoted to have made a dissatisfactory statement.
"Some people think they are tall but they can never control us. Never!", the unidentified governor was quoted as saying.
The governors were reportedly engaged in fierce arguments during the meeting, hence President Buhari asked them to pick another date to meet again.
Governor of Zamfara state Abdulaziz Yari was said to have chosen April 9 but el-Rufai reportedly opposed the date.
It is not clear as at the time of filing this report if the April 9 date was eventually agreed upon by all the governors.