Jammeh, in power for 22 years, had accepted defeat a day after the vote, but backtracked a week later
Nigerian President Muhammadu Buhari, Liberian leader Ellen Johnson Sirleaf along with the presidents of Sierra Leone, Ghana and the UN envoy for West Africa, Mohamed Ibn Chambas, are set to arrive in Banjul on Tuesday.
Council members "urge President Jammeh to meet and cooperate with the UN-ECOWAS high-level delegation that will visit Banjul tomorrow," said Spanish Deputy Ambassador Juan Manuel Gonzalez de Linares.
Spain holds the council presidency this month.
The council met behind closed doors at Senegal's request to discuss the crisis over Jammeh's refusal to recognize the outcome of the December 1 election.
Jammeh, in power for 22 years, had accepted defeat a day after the vote, but backtracked a week later, saying he rejected the outcome and calling for a new vote.
US Ambassador Samantha Power said it was a "very dangerous moment" for the West African country, citing reports that some military officers have sided with Jammeh in the standoff.
The Security Council on Saturday released a unanimous statement demanding that Jammeh "transfer, without condition and undue delay, power to President-elect Adama Barrow."
Gonzales said the council "reiterated the full contents" of that statement during their meeting on Monday and called for maximum restraint.
The deputy ambassador said there was no specific discussion of measures to force Jammeh to step down.
"In this moment, our maximum priority is to support the high-level visit that is going to take place tomorrow in Banjul," he said.
The envoys will convey a message to Jammeh on behalf of the African leaders and the United Nations, he said.
Power stressed that Jammeh "has the chance to establish a very different kind of legacy" and that the high-level delegation will send that message if they meet with him on Tuesday.