The Gunners have not added any silverware since 2005 and hopes of an unexpected Premier League title triumph drifted away in the closing months of the campaign.
The 33-year-old - who has captained the Blues 490 times during his career - is understood to have originally been negotiating for a longer deal, but has had to accept the west London club's policy for players over the age of 30.
Vidic, who is joining Inter Milan this summer, insists former manager David Moyes was not solely to blame, and revealed the players argued amongst themselves more than any other year in his career at Old Trafford.
The former Spurs midfielder sympathised with the way in which he was dealt with by Daniel Levy, which resulted in a departure from White Hart Lane just five months after being appointed as Andre Villas-Boas's successor.
Rodgers snapped up eight players last summer, but he does not anticipate it being such a busy time at Anfield after the Reds finished second in the Premier League.