The Switzerland international was spotted in the stands at the Britannia stadium during Stoke’s clash with Liverpool.
The report states that Everton boss is a keen admirer of the Inter Milan winger however, Stoke are understood to be ahead in the race having agreed a £13.5 million deal with Inter earlier this summer.
Galatasaray are also in the hunt for the 23-year-old according to the Turkish newspaper Fotomac.
The daily sports newspaper claim Galatasaray will hold talks with Inter Milan next week.
Shaqiri who joined Inter for £11.2 million from Bayern Munich last season provided four assists and scored five goals in 35 appearances for the Serie A giants.
The former Basel player has scored 17 goals in 46 appearances for the Swiss national team aged just 23. Shaqiri has four years remaining on his contract with Inter Milan.