Bundesliga strugglers Borussia Dortmund have agreed to sign Slovenia international Kevin Kampl in January.
Cardiff have agreed to sell Norwegian midfielder Mats Daehli to Freiburg in January.
Phillips has been a free agent since last weekend after his contract expired with Crystal Palace, with whom he won his fourth promotion to the Barclays Premier League last season.
He scored the match-winning penalty against Watford in their Sky Bet Championship play-off final at Wembley, but made just four appearances from the bench for the Eagles in the top flight.
Plenty of clubs were interested in signing the 40-year-old, but the former Aston Villa, West Brom and Birmingham striker said he wanted a return to the Midlands after leaving Selhurst Park and Championship leaders Leicester have won the race for his signature.
Foxes boss Nigel Pearson, who worked with Phillips at West Brom, confirmed his interest in the player following his side's 4-1 league win over Derby at the King Power Stadium last Friday.
After completing the deal Pearson said: "The opportunity to sign a player of Kevin's experience and know-how doesn't come along too often so I'm delighted that we've been able to add him to our squad.
"He's a player I know well, he's a fantastic professional with multiple promotions to his name and he will bring some great qualities to the group, both on and off the pitch."
Former England international Phillips began his career with non-league Baldock Town and also went on to play for Watford, Sunderland, Southampton and Blackpool during a 20-year career in which he has scored a total of 285 goals in over 650 appearances.