On Saturday they take on Worcester Warriors in Round 3 at Sixways Stadium as Pool 2 leaders, with their back-to-back contests against the English Premiership side set to go some way to determining quarter-final qualification.
Connacht coach Michael Bradley said: "We are in pole position in the group as we have won both our games but our next three Challenge Cup matches are against Worcester, Worcester and then Montpellier, so after those games we will know exactly where we are in regard to qualification.
"If we can gain the upper hand in those back-to-backs against Worcester then we will be well placed.
"The change in the rules for quarter-final qualification so that only the pool winners go through, has certainly made things much more difficult.
"But if we can top our group and qualify that would be a huge progressive step-up for us and something that would be a real feather in our cap."
Connacht will go into the Sixways showdown on the back of a 21-9 Magners League defeat against Cardiff Blues with Bradley relishing the European resumption.
"Reaching the Amlin quarters would be a great achievement and big part of our season," added the former Ireland scrum half.
"We always enjoy playing in the tournament, travelling to different places and coming up against foreign opposition and testing ourselves against quality opposition, which Worcester and Montpellier most certainly are.
"Along with ourselves that makes for three relatively even sides and, while we are always competitive, we also know that, on our day and if things go for us, we can win those evenly balanced matches.
"We have great respect for Worcester and the match on Saturday at Sixways promises to be a real challenge but we also know the opportunity is there for us.
"We played Worcester in the tournament a few seasons ago and we know from those games that they certainly fight very hard and since then they have added a number of overseas players who have brought pace and speed to their back-line.
"If they can establish a strong forward platform then they can be a very dangerous side.
"The result in Cardiff at the weekend means we have now won four from nine and overall I am happy in terms of where we are.
"And I am delighted that our captain John Muldoon and Ian Keatley got capped by Ireland during the summer and now Sean Cronin has joined them during the autumn internationals.
"We also had several players representing Ireland 'A' so we are going on along quite nicely."
On another positive front, new signing George Naoupu has been declared fit to face Worcester. The Kiwi back rower agreed to join the province from Super 14 outfit the Highlanders at the end of last month.
A former New Zealand Under-21 star, the 26-year-old is in line to make his first appearance in Connacht colours after signing on until the end of the season.