Budaj has been impressive this season as the second crease option behind Carey Price. The Slovak goalie is 6-1-1, with one shutout, a 2.17 goals-against average and .917 save percentage.
The club did not release financial details. Budaj is in the last year of his current contract, with a prorated cap hit of $1.15 million US per season.
The 30-year-old signed as a free agent prior to the 2011-12 season after spending six seasons with Colorado.
For his career, Budaj has 10 shutouts, a 2.79 average and .903 save percentage in 267 career regular season appearances.
According to reports, Budaj is the likely starter when the Canadiens play in Buffalo on Thursday in a Hockey Night in Canada contest (CBC, CBCSports.ca, 7:30 p.m. ET