With two tests still to play, the former Zimbabwe international faces the possibility of an embarrassing 5-0 whitewash.
"We believe we've got one of the best men in world cricket. We hope Andy will take us forward and into the future and the 2015 Ashes," Collier told the BBC.
"Andy has the total and full confidence of the board. He has done an outstanding job," he added. "We are very hopeful he will remain with us. We believe he is the best man for the role."
Flower, 45, has received widespread support among England's players, with bowler Steven Finn and batsman Joe Root echoing Collier's backing.
"He's been fantastic for the group," Finn told BBC radio. "He's always working extremely hard to make sure that we're as prepared as we can possibly be for test matches. It would be a great loss if he goes."
Root, who was given his debut by Flower in 2012, said: "He has done a fantastic thing for English cricket over his career. It's been great working with him and hopefully that can continue."
Flower has delayed making a decision on his future until after the current tour.
Off-spinner Graeme Swann compounded England's woes by announcing his immediate international retirement on Sunday.
The 34-year-old appeared in 60 tests, taking 255 wickets, making him the sixth highest England wicket taker.
Yet Swann has toiled in Australia taking only seven wickets in the opening three tests.
"Graeme Swann has made an outstanding contribution to the England cricket team in all formats throughout an incredibly successful career and I would like to congratulate him on all that he has achieved," Flower said on the ECB website (www.ecb.co.uk).
"His commitment, competitive spirit and sense of humour have been recognised and admired by team mates and supporters alike and he has played a big part in England's success over the last five years."
Former England captain Michael Vaughan praised his impact and said he will be very hard to replace.
"Of the last 20 years, Swann is going to be one of the hardest to replace," said Vaughan.
"England without Swann in the last five years, I just can't see how many series they would have won."
Former England bowler Derek Pringle, however, criticised the timing of Swann's announcement and said he "should have seen the tour out as a senior player".
(Reporting by Tom Hayward; editing by Toby Davis)
