"He is due back tomorrow," Blue Jays manager John Farrell said. "Right now [in time for game], yes."
Clubs may place a player on the paternity list for a minimum of one day and a maximum of three days. The roster move cleared the way for left-hander Evan Crawford to be recalled from Double-A New Hampshire, which now gives the Blue Jays eight relievers in their bullpen.
During Santos' absence, setup man Francisco Cordero will take over the ninth-inning duties.
Santos got off to a rough start this season by failing to convert his first two save opportunities. He bounced back on Wednesday by striking out Boston's Kevin Youkilis and getting David Ortiz to ground out to preserve Ricky Romero's first win of the season.
The right-hander has allowed four earned runs in three innings this season. He has a 3.51 ERA in 118 innings with the White Sox and Blue Jays.
Crawford hasn't allowed an earned run in three appearances for Double-A New Hampshire this season. He was a ninth-round selection in the 2008 First-Year Player Draft and made a strong impression in Spring Training with a fastball that tops out at 94 mph, along with a very effective curveball.