The merger agreement was previously scheduled to be completed on October 27.
The companies entered into a definitive agreement in October 2015, under which Walgreens would acquire all outstanding shares of Rite Aid for $9 a share. The original terms of the agreement allowed either party to delay the merger until January.
Walgreens Boots Alliance is the largest retail pharmacy chain in the U.S. and Europe. Walgreens' stock was trading more than 4 percent higher Thursday.
Shares of Walgreens Boots Alliance and Rite Aid have fallen more than 4 percent and 9 percent year-to-date, respectively, according to FactSet.