Bild reports that the 53-year-old arrived at Borussia-Park on Monday morning and is expected to sign a three-year contract shortly.
Gladbach have reached an agreement with St. Pauli to release him from his existing deal, although the details of that arrangement have not been made public, and Fussballtransfers.com says he has in effect been settled with the bottom club for some time.
Blessin was relieved of his duties in Hamburg at the start of June, but he was still under contract until 2027. He coached St. Pauli for two years, keeping them up in 2024/25 as a newly promoted side before they were relegated last season after a home defeat to Wolfsburg on the final day.
He takes over a five-time German champion in crisis. Gladbach have lost all four of their Bundesliga games, are the only side without a point, and have conceded 16 goals, which is the worst start in the club's history.
Polanski was dismissed after only three matchdays, having lost each of them, following a 5-1 defeat at Freiburg, and Jan-Moritz Lichte was in charge as interim coach for last weekend's 3-4 home defeat to Mainz, in which Gladbach conceded four goals again.
Sporting director Rouven Schröder had said after that Mainz game that the club had reached a verbal agreement with a coach, without naming him, and that they would use the coming days to make it official.
The timing helps Blessin. He starts his new job during the three-week international break, which gives him time to settle in and work with the players who remain at the club before his first match. That is a Rhine derby on 11 October away to 1. FC Köln, with a 15:30 kick-off, an immediate test against a rival.
A three-year contract would run until 2029, which suggests Borussia Monchengladbach see him as a longer-term solution rather than a short-term fix. The first priority, though, is to earn points and to stabilise a defence that has been breached 16 times in only four games.