Lancaster has revitalised a dispirited England set-up in just three months and led a new-look side to four wins from five in the Six Nations earlier this month.
Rookie Lancaster, who has never coached a top club, faces competition from decorated former South Africa and Italy boss Nick Mallett.
The Rugby Football Union is expected to make an announcement imminently – and Dowson has left them in no doubt which man the players want.
“If you asked the whole squad, to a man they would say they’ve got faith in the coaches,” said the Northampton forward. “The proof is in the results. I think all the players would be disappointed if it wasn’t Stuart Lancaster.”
Previously coach of England’s second-string Saxons team, Lancaster, 42, promoted a number of his protegees with great success and won praise for fostering a strong collective mentality.
Dowson added: “There are a lot of psychological things he did – meetings during training, before games and after games – and I think the players responded to that and I think the results were borne out of that.”
Leicester fly-half Toby Flood, who did not feature heavily due to fitness concerns and the form of young Saracens back Owen Farrell, praised Lancaster’s “excellent job”.
“He created an environment where all the boys were keen to work hard for each other,” Flood added. “If you take into account what he has managed to do in such a short space of time then he has to get the job.”
Flood conceded, however, that new RFU chief executive Ian Ritchie had a “difficult” choice to make. “The RFU have to take all sorts of things into account such as experience and knowledge of players, things that we do not know about as players,” he said. “But Stuart certainly did himself no harm at all.”