Barack Obama's special envoy to central Africa has warned an armed group in the Democratic Republic of Congo (DRC) that it has months to surrender or face "the military option".

Speaking at the US president's summit with African leaders in Washington, Russell Feingold said there was a "tremendous need to finish off" the Democratic Forces for the Liberation of Rwanda (FDLR), a group formed by the leaders of the genocide in Rwanda 20 years ago and which controls parts of eastern DRC.

"We have to get rid of the FDLR, not so much because of their military capacity, but because of what they represent and the destabilising effect that they have with regard to relations with Rwanda. That is our top priority," he said. "I've been involved with efforts to communicate to them that it's time for them to surrender. That they will be attacked militarily if they don't. That there will be no political dialogue."

Feingold noted that another rebel group, M23, which was backed by Rwanda and carried out massacres, rapes and other war crimes in DRC, was forced to surrender last year by a combination of diplomatic pressure and military action by the UN and African forces. It is likely that the FDLR would face a similar military operation if attempts to pressure it into surrendering fail.

The FDLR was formed by leaders of the genocide of about 800,000 Tutsis in Rwanda. Former soldiers and militias that carried out the killings fled in to what was then Zaire in 1994 after the defeat of the extremist Hutu government in Kigali.

For several years, its forces led cross-border raids into Rwanda, usually killing civilians. The assaults, however, fell away and the FDLR settled in to control an area of eastern DRC close to the Rwandan border, where it fought government forces and attacked the local population, killing hundreds and possibly thousands of people. Human rights groups have also accused its members of mass rape.

The FDLR is no longer regarded as a serious military threat to Rwanda, but it has kept "genocide ideology" alive as it raised a new generation to hate Tutsis.

Much of the leadership lives abroad, some of them in Europe. The FDLR chairman, Ignace Murwanashyaka, and his deputy, Straton Musoni, were arrested in Germany and tried for war crimes and crimes against humanity. A verdict is expect later this year. The group's executive secretary, Callixte Mbarushimana, was detained in France and extradited in 2011 under an international criminal court warrant. He was released later that year on the grounds that there was insufficient evidence to prosecute him.

Feingold's warning to the FDLR follows a regional agreement to end the activities of all armed groups in eastern DRC, some of which have been backed by Rwanda and Uganda, and to return control of the region to the government in Kinshasa after two decades of foreign invasions, civil war and plunder of its mineral resources. Many of the armed groups have been responsible for mass rape and indiscriminate killings.

The FDLR leadership initially indicated that it would surrender but then demanded political dialogue with the Rwandan government of President Paul Kagame, who led the then rebel army, which defeated the extremist Hutu regime. Kagame has refused to negotiate.

Feingold said that mediators were prepared to facilitate the return of FDLR members to Rwanda or to settle elsewhere, presumably in DRC, but that there would be no talks about it taking a political role in Rwanda – principally because of its ties to the genocide.

Several previous attempts to disarm the FDLR have failed, including a cross-border assault by the Rwandan army. The surrender of M23, however, indicates a determination by the UN and regional powers to finally put an end to violence in eastern DRC.

Feingold said that once the FDLR was dealt with, the key to long-term stability in DRC would be to strengthen the political system. That includes the country's president, Joseph Kabila, stepping down at elections in 2016 as required by the constitution. He said the US secretary of state, John Kerry, told Kabila at a meeting on Monday that Washington expects him to abide by the two-term limit and not to attempt to hold on to power.

"We regard the elections as one of the top priorities not only for the domestic DRC, but for the stability of the entire region because without a credible political system it is unlikely that the eastern DRC will achieve the stability and the kind of sovereignty over the area that is needed," he said.