Czech TV Nova said both sides would sign the new treaty on April 8 in Prague, where Obama gave a major speech last April calling for a world free of nuclear weapons while acknowledging he may not live to see that goal achieved.
Obama and Medvedev set a goal in July of slashing the number of warheads on either side to between 1,500 and 1,675 warheads and the number of "carriers" capable of delivering them to between 500 and 1,100.
The US has said it currently has about 2,200 nuclear warheads, while Russia is believed to have about 3,000.
Moscow and Washington have held months of difficult negotiations in Geneva aimed at replacing the treaty, a cornerstone of Cold War-era strategic arms control.
Signed in 1991, Start led to huge reductions in the Russian and US nuclear arsenals and imposed verification measures to build trust between the two former Cold War foes.
Delays in the Start talks and missed deadlines have cast a shadow over the Russian and US leaders' efforts to make good on their pledge to improve bilateral ties.
The US is set to host a nuclear security summit on April 12 to 13, and observers have said it is a matter of pride for Washington to have the new treaty in place before the summit.
A review conference for the Nuclear Non-Proliferation Treaty is scheduled for May.
Analysts said US Republicans who staunchly back missile defence may try to deny the Obama administration the two-thirds majority it needs in the Senate to ratify the Start treaty.