Each side has won one and lost one of their opening two fixtures, turning the Twickenham collision into a must-win encounter if they are to remain in title contention.
“It’s a big game and we’re treating it like a quarter-final. We have to target three wins. It’s knockout stages for us. If we don’t get the result on Saturday, we’re out,” said Gleeson.
“We know Wales fight really hard and will make it really difficult for us, so we’ve got to go into this with a quarter-final mentality and go after Wales. We’re going after them.”
England enter the round three clash with an almost fully fit squad, with only uncapped Bath prospect Orlando Bailey unavailable because of a hamstring strain.
All other members of the 35-strong group assembled at their Surrey base came through Tuesday’s training session, giving head coach Eddie Jones options in selection for Thursday’s team announcement.
Courtney Lawes is on course to make his return as his comeback from concussion gathers pace and Jones faces a decision on whether to reinstate the Lions flanker as captain.
Tom Curry deputised against Scotland and Italy in Lawes’ absence and England have been impressed with the maturity he has shown in the role.
“Tom’s led with his actions. His stint against Italy was immense. He was all over the place. He broke records for his GPS for the amount of work he got through,” continued Gleeson.
“He’s been really good for us and whether he’s captain or not, he’ll continue to be a pivotal part of the team.
“Courtney’s been brilliant. He started training again a little while ago and he’s been ramping it up slowly.
“He was back out there today (Tuesday) doing what he does - leading by example. It was great to see.”
The biggest lift is supplied by Manu Tuilagi as the bulldozing Sale centre continues to prove his fitness after three months out with a torn hamstring.
Tuilagi’s availability comes just as England face make or break in their Six Nations title quest and he is likely to be restored to the centres alongside Henry Slade, giving the midfield the balance it lacked in rounds one and two.
“Manu adds lot - his physical presence, his footwork, his leg speed. The things he can do, not many other people can do. That gives you a new dimension to play off. It’s great having him back in the mix for selection,” added Gleeson.
“He gives us that little bit of flexibility in what we can do. Sladey and Manu played really well together in the autumn and have done previously.
“If both are selected this weekend they’ll continue to do so. It will be interesting if they both take the field.”
Meanwhile, Wales prop Tomas Francis is ready to ignore Ellis Genge’s “mind games” in Saturday’s clash at Twickenham.
The pair are set for a head-to-head scrummaging battle after both countries revived title aspirations with victories last time out, with England beating Italy and Wales toppling Scotland.
Francis is among only a handful of the current squad to have won with Wales at Twickenham, collecting his fourth cap during a memorable 2015 World Cup pool stage encounter.
England prop and Leicester captain Genge is a no-nonsense operator and a pivotal member of England’s pack.
Asked what he was expecting from Genge and his fellow England loosehead Joe Marler, former Exeter forward Francis said: “From Genge, you have always got those mind games.
“He likes to try and get into you that way. I try not to engage in it.
“It’s just part of his game. It’s something (England and Exeter lock) Jonny Hill told me once. I just focus on me being me, and him being him, to be honest.
“He has got that leadership role at Leicester, and he seems to be thriving in that.
“His performances on the pitch have grown as well. He has become a world-class loosehead over the years, and he is doing well. Marler has had a resurgence, and he is also playing well.
Whoever is there is going to be a big challenge, and they are a tough pack to go up against.
“Whoever plays for Wales has got to have that mindset that you are going to go out there and dominate – at least parity, but try and dominate and give a good platform for the backs.
“England will be a great test, they have got some great forwards, and hopefully we can come on top. We’ll give it our all.”
Wales head coach Wayne Pivac will name his team on Thursday, with his biggest decisions set to revolve around midfield and whether he fields fit-again number eight Taulupe Faletau, who was called into the squad this week. Backs Josh Adams and Uilisi Halaholo, meanwhile, are both available after missing the Scotland game through injury.
Wales have not won a Six Nations game at Twickenham since centre Scott Williams’ breakaway try secured a 19-12 success 10 years ago.