Adrian first joined West Ham in 2013, while the club was under the management of Sam Allardyce.
The Reds would therefore be able to secure his services without paying a transfer fee.
The 32-year-old made 125 Premier League appearances for West Ham during his time in the East End.
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Mignolet travelled straight from Wembley following the Community Shield fixture against Manchester City. That same evening, Club Brugge announced they had signed "one of the best goalkeepers in the Premier League".
Adrian was a free agent after leaving West Ham when his contract expired following six years in the capital and he chose not to extend his stay at the London Stadium after losing his place to Lukasz Fabianski.
Adrian will be Klopp's third signing of the summer, having spent just £1.7m.
If Adrian does arrive he would join teenagers Sepp van den Berg and Harvey Elliott in signing for the European champions this summer.
Mignolet played an important role day-to-day in terms of putting pressure on Alisson and his team-mates recognise that Mignolet has never allowed his standards to drop or mood to change.