Bailey had right shoulder surgery in July before missing the second half of the season, which ended with Boston winning the World Series.
The righty earned $4.1 million in 2013, which means the Red Sox must be fairly confident he will make a full recovery if they tender him a contract.
Any player who is non-tendered by the deadline becomes a free agent.
Lefties Andrew Miller and Franklin Morales also must be offered contracts by Monday.
Miller excelled last season before sustaining a severe left ankle injury just before the All-Star break. He made a modest $1.475 million this year, and it seems all but certain he will be back.
Though Morales had a hard time staying healthy, his versatility makes him a valuable member of the team.
Righty Junichi Tazawa, a star in this year's postseason, is another no brainer for Boston to re-sign.
Ditto for first baseman/outfielder Mike Carp, who did not have a set role on last year's team but seemed to come through every time he got the opportunity. The left-handed hitter had a .296 average with nine homers, 43 RBIs and an .885 OPS in 216 at-bats.
Right-hander Burke Badenhop, acquired recently from the Milwaukee Brewers, is also arbitration-eligible.
Ian Browne is a reporter for MLB.com. Read his blog, Brownie Points, and follow him on Twitter @IanMBrowne. This story was not subject to the approval of Major League Baseball or its clubs.