Steve Coppell resigned from his post with the Royals following their play-off semi-final defeat to Burnley, but the Hornets' chief executive Julian Winter insists "there has been no contract at all" from Reading.
Rodgers remains focused on the task at hand at Vicarage Road and told the Watford Observer: "When I am asked about other clubs, people are questioning my integrity and one thing I have mentioned is I always have integrity.
"I am loyal and find it disloyal when I am asked about other cubs when I am the Watford manager. There is nothing that has changed in that respect.
"I have seen the papers and know I am favourite for the job but that will always happen in football and, in fact, my job was linked with someone else this year.
"I cannot let that speculation hamper my focus or objectives. It doesn't concern me. What concerns me is that I am the manager here and that has been my focus right the way through."