Sergei Lavrov's comments were Moscow's strongest public statement yet that a deal may be imminent.
"The remaining questions, I hope, will be resolved rather promptly when the negotiations resume, and they will resume at the very beginning of February, I think," Lavrov told reporters.
U.S. President Barack Obama and Russian President Dmitry Medvedev laid out plans last year to forge a successor to the Strategic Arms Reduction Treaty and reduce the arsenals of the two largest nuclear powers.
Negotiators were unable to reach agreement by December 5, when START I expired, and official negotiations in Geneva have not resumed after a break over the holiday period. A top U.S. official had indicated earlier this month that they would resume on January 25.
But high-level consultations on the treaty resumed last week, and two top U.S. officials, national security adviser James Jones and chairman of the Joint Chiefs of Staff, Admiral Mike Mullen, travelled to Moscow this week for talks.
A spokesman for Jones said the retired general and Mullen held talks with Medvedev on Thursday on the START negotiations and also about the way forward in the stand-off with Iran over its nuclear program.
"What you will see in the coming weeks is our negotiating teams coming together to finalize the details. But we are certainly optimistic that the Start agreement is within reach," spokesman Mike Hammer told reporters in Washington.
Lavrov said Jones and his Russian counterpart were expected to give the negotiators instructions that would help reach compromises. He did not say what remains in dispute or precisely when a final agreement might be reached.
Both sides have said they want the treaty signed in time to set an example for a global conference in May that they hope will bolster efforts to combat nuclear weapons proliferation.
Any agreement must be ratified by lawmakers in both countries to take effect.
In July, Obama and Medvedev agreed that the new treaty should cut the number of nuclear warheads on each side to between 1,500 and 1,675, and the number of delivery vehicles to between 500 and 1,100.
Analysts say negotiators are at least closer to agreement on more specific numerical limits within those ranges.
Officials have said recently that issues still being negotiated included monitoring and verification measures.