"This is an incredibly important week for Greece. The level of anxiety could run high but there is still optimism there will be a resolution between Greece and its creditors," said Ed Atkins, Treasury strategist at RBS Securities in Stamford, Connecticut.
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Tensions between the new Greek government and other members of the euro zone on the implementation of economic reforms in order to obtain aid have raised the possibility of the country exiting the economic bloc, a move that traders fear could roil financial markets.
Greek Prime Minister Alexis Tsipras and German Chancellor Angela Merkel were set to meet for the first time later Monday as Tsipras seeks support from the head of the euro zone's most powerful member.
Benchmark 10-year Treasury note were yielding 1.91 percent, down 2 basis points from late on Friday.
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The yield spread between 10-year Treasurys and 10-year German Bunds narrowed to 1.73 percent from 1.74 percent on Friday as German yields retraced a tad from last week's record lows stemming from the European Central Bank's 1.1 trillion euro bond purchase program.
The rock-bottom yields across Europe have spurred overseas demand for higher-yielding U.S. debt.
This week's Treasurys fixed-rate supply will kick off with a $26 billion auction of two-year notes on Tuesday, followed by $35 billion of five-year debt on Wednesday and $29 billion of seven-year notes on Thursday.