The 32-year-old midfielder is out of contract with Schalke at the end of June and has failed to agree a new deal.
Schalke reportedly offered Caligiuri only half of his currently €2.7m annual salary for a new signature.
And BILD reports that the veteran is in advanced talks to join Augsburg, who are currently three places below Schalke in the Bundesliga.
The former Wolfsburg star has been with Schalke for three-and-a-half seasons, making 125 appearances in all competitions.
Caligiuri would be another important building block for Augsburg, who have already secured Tobias Strobl from Borussia Mönchengladbach.
The Bavarian club have also made Felix Uduokhai’s loan move from Wolfsburg permanent.
In addition, Augsburg are also trying to sign Swiss right-back Kevin Ruegg but BLICK says no deal is close.