John Stremlau, the Carter Centre's vice-president for Peace Programs, said the former president would take such a request for all parties seriously.
However, he said it was "woefully premature to suggest [Carter] will plunge himself into mediating this conflict".
Output from Africa's largest oil producer has been cut by around a fifth because of attacks on the industry by MEND.
The group publicly approached Carter earlier this year to act as a negotiator.
"We are ready to call off all hostilities and hold a temporary ceasefire in honour of president Carter should the Nigerian government accept president Carter's initiative," MEND said in the statement.
"However, if as expected, the government fails to seize on this new opportunity for peace, our actions will continue to speak volumes beyond the Nigerian shores."
Carter failed in a previous attempt to mediate in the restive Delta in 1999, but he is familiar to many senior members of the different armed factions.
The Carter Centre informed MEND on Monday that it would consider mediating in the long-running conflict only if all sides made the request.
"The absolute first condition (for mediation) is that the Nigerian government has to approach us," Stremlau told Reuters news agency.
Since it rose to prominence two years ago, MEND has carried out the vast majority of the large-scale attacks on foreign oil companies operating in the south of Nigeria, the world's eight crude exporter.
Such attacks, together with operations by local communities, have cut Nigeria's production by about a quarter over the past two years.
Through his centre Jimmy Carter, president from 1976-1980, has specialised in election observation and mediations throughout the world.
Carter recently met top Hamas officials in Syria, despite objections from Israel and criticism from the Bush administration.