The ban would be a reaction to the U.S. decision to suspend military cooperation with Russia over Ukraine, the source, who requested anonymity, said Saturday.
Russia has vehemently opposed Ukraine's pro-Western government that was installed last month following street clashes between police and the opposition led by far-right nationalist groups.
Russia was also accused of backing separatists in the Ukrainian republic of Crimea, which is planning to secede, and deploying thousands of troops there, though Moscow has denied sending troops to the pro-Russian region.
The New START, signed in 2010 by U.S. President Barack Obama and his Russian then-counterpart Dmitry Medvedev, envisages curbing nuclear arsenals of both countries by half by 2021.
The treaty's implementation is overseen, in addition to satellite and remote monitoring, by 18 on-site inspections a year.
The U.S. has not conducted any inspections under the New START since the start of this year, according to the U.S. Department of State's website.