Spurs boss Mauricio Pochettino signed left-back Davies in a £10m deal from Swansea City this summer, but Rose has started all three of Tottenham's Premier League outings so far this campaign.
"This competition is going to help us both," Davies told the club's official website.
"It is going to make us both better players, hopefully, and that's all we want to do as professionals. Danny has been great to me and it is a competition between us.
"We both want that jersey, but whoever does get it at the end of the day we are supporting the team and we'll support each other. That's the way it is going to be."
Tottenham travel to the Stadium of Light to take on Sunderland on Saturday.