The moves will help off-set Stephane Sessegnon's move to West Brom which was completed as the end of the transfer window approached.
The 22-year-old Borini was Liverpool boss Brendan Rodgers' first signing at Anfield when he joined for around £10million from Roma last summer, but struggled to hold down a regular first-team place.
Borini has been capped 18 times by Italy at Under-21 level and made his full international debut in a friendly against the United States in February last year.
Borini will provide extra attacking options for Di Canio and fill the gap created by the departure of Sessegnon.
Borini joined Bologna as a youth player before moving to Chelsea in 2007. After a loan spell at Swansea he moved back to Serie A for brief periods at Parma and Roma.
Meanwhile Dossena joined Liverpool from Udinese in 2008 but after losing his first-team place moved back to Serie A with Napoli 18 months later.
The 31-year-old, who has 10 senior international caps, helped Napoli win the Coppa Italia in 2012 with a 2-0 win over Juventus in Rome.
The double signing takes Di Canio's Italian haul to four after previous deals for midfielder Emmanuele and goalkeeper Vito Mannone.
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- Stephane Sessegnon