The two governors, who are among the newly elected federal lawmakers currently attending an orientation programme in Abuja, excused themselves from the venue and had a discussion for a long time.
Concise News understands that their long talk was connected with the issue of the Senate presidency.
Amosun had declined comments on Monday when journalists asked him to confirm or deny the reports that serving and former governors were plotting to install the 9th Senate President.
It was said that his meeting with Shettima on Tuesday might not be unconnected with the decision of the All Progressives Congress to adopt a candidate for the position.
Their body language when they saw an aspirant to the position, Senator Ali Ndume, also fuelled the suspicion as two governors also engaged the former Senate leader in a brief discussion before resuming their talks.
The governors returned to the venue of the programme after they had discussed for over one hour.