Southampton however, are not alone in their pursuit of the England U21 starlet, who has scored four goals in eleven appearances for the U21 side.
The Saints are keen to add Redmond to their squad during pre-season, with Ronald Koeman rebuilding the side in the aftermath of the departures of Luke Shaw, Adam Lallana and Rickie Lambert - with Dejan Lovren expected to follow suit with an exit in the near future.
If the former Birmingham City player was to sign for Southampton, the move would see him join up with England U21 team-mate and captain, James Ward-Prowse.
Norwich City are reported to want a figure closer to £6m before they would consider allowing Redmond to move on, but Neil Adams is keen to ensure that the winger remains at Carrow Road for the foreseeable future.
Robert Snodgrass has already left Norwich City this summer, in an £8m switch to Hull City and Norwich are keen to keep departures down to a minimum as they aim for an immediate return to the Premier League.
Southampton are looking to secure their third signing of the summer, after recruiting Eredivisie duo Graziano Pellè and Dušan Tadić from boss Koeman's homeland.
Stoke City have already allowed Matthew Etherington and Michael Kightly leave the Britannia Stadium, creating room for a new wide attacking option, following the arrival of Bojan Krkic from FC Barcelona on Tuesday.