The 22-year-old trained last week with 2013 FA Cup winners Wigan Athletic after a number of encouraging displays for the Vanarama National League club.
Birmingham City have also been in touch with Sutton, who has a good recent history of helping players to progress.
Doswell says there are a number of players good enough to play higher up the pyramid in his squad and that the U's will continue to provide a shop window for talent.
"Josh has been the subject of interest from a number of clubs including Birmingham City and Wigan Athletic," the manager confirmed.
"He trained with Wigan on Monday and Tuesday last week and did very well, and has been invited back when our schedules suit. Josh has been superb since he joined and his work ethic and personality in the group has shone through.
"As a club we take huge satisfaction in letting our young players progress into the Football League and this model is vital in attracting these players in the first instance to Sutton United," he told the club's website.
"We currently have four first team players in Anthony Jeffrey, Louis John, Tommy Wright and Josh Taylor who are 23 or under and the interest in all these players shows how much progress they are making.
"Louis in particular has grown into a really good centre half and fully deserved his England C call up. Tommy Wright has also shown huge improvement, and it's a great credit to him and his work on the training ground that he has started the season so well, and kept more senior players out of the side.
"These young players have a fantastic environment to work in and great credit should go to the senior players and coaching staff who have passed on such valuable knowledge and encouragement to them."