The bids were made on Wednesday night but they fell short of the £27m which Arsenal quoted to Crystal Palace for the 22-year-old earlier this month.
The Gunners had been happy for Chambers to leave after telling him he would not be guaranteed a place at centre-back.
However, they have since changed their mind when Arsene Wenger decided he wanted the England international to stay.
Chambers joined Arsenal in 2014 but spent last season on loan at Middlesbrough, who were relegated at the end of the season.