Fletcher's men looked down and out after losing twice in the competition before Christmas, but successive victories over French challengers Montauban and Brive mean a bonus point success against Italian minnows Petrarca Padova on Saturday might be enough.
Guinness Premiership clubs could provide half the quarter-final contenders, with Saracens already through, Bath and Bristol likely to follow suit, while Newcastle remain well-placed in Pool Three.
Fletcher said: "After bonus point wins against Montauban and Brive, we are pretty positive about the situation.
"Our destiny is in our own hands now, barring Harlequins scoring an absolute barrel-load of tries against Connacht.
"We could even win our group if Brive don't get anything at home to Montauban, although in all honesty I would expect Brive to come out on top in that one."
Newcastle will line up at Stadio Plebiscito with England centre Mathew Tait in the unaccustomed position of full-back, but Fletcher added: "Mathew has characteristics I feel are very well suited to that role.
"I think he will be a sensational full-back, and it was always part of my planning to give him a run there at some point.
"This thing about players playing only one set position is absolute nonsense to me. There is less than 10% of the game where everyone has to line up in a certain place, and the rest of it is just phase play."
Harlequins, currently five behind Newcastle on try-count, face an uphill struggle to progress from Pool Four and are likely to pay a hefty price for back-to-back defeats against Bath last month.
Quins rugby director Dean Richards said: "Losing both games to Bath has put us out of the running in terms of winning the pool, but we are still in it in terms of qualification for the quarter-finals.
"We have had some very good results in recent weeks, putting in some strong and convincing performances which have given us a real lift."
If Quins and Newcastle both claim maximum points on Saturday - Bath, meanwhile, should win their group by crushing Recreation Ground visitors Montpellier - then it could spell the end of Premiership basement club Worcester's European campaign.
The Warriors can finish with a total of 20 points in Pool Five, but that might not be sufficient, even if they run Italian hosts Viadana ragged on Saturday.
Worcester rugby director John Brain has made eight changes from the side beaten by tournament favourites Clermont Auvergne, with prop Callum Black, lock Will Bowley and fly-half Simon Whatling gaining chances, while 19-year-old prop Matt Mullan is on replacement duty.
Brain said: "Looking at the fixtures, it is going to be extremely difficult for us to make the quarter-finals.
"We would like to win and have a bit of a feel-good factor going into the Wasps (Premiership) match next Friday.
"The objective is to rest one or two players, while others are coming into form, and we want to develop that."