While Llanelli Scarlets and The Dragons contest the title with Ulster, Celtic Warriors (48 points) are in the thick of chasing one of the two Welsh places with Neath-Swansea Ospreys (45 points) and Cardiff Blues (42 points).
"Our No 1 priority is to qualify for Europe, there is no hidden agenda for the Warriors," said coach Lynn Howells. "Making sure we are in the Heineken Cup next season is our only target over the remainder of the season.
"We do have a six-point cushion at the moment but, with 20 points still up for grabs over the final four rounds of matches, I honestly think it is going to go down to the wire - both in the chase for the Celtic League title and in the battle for Heineken Cup qualification.
"We have four matches left and we have to focus on doing everything we possibly can to keep getting the results - starting with this Friday night.
"I would have thought Glasgow Rugby would have much the same attitude as us, getting into a Heineken Cup spot, and we are expecting a tough battle at Hughenden.
"When we played them earlier in the tournament it was a real dogfight and we are expecting something similar this time.
"We found them very difficult to play against and their forwards certainly gave us plenty to think about.
"We will be without Jon Bryant, who has a hamstring injury, but otherwise we should be able to select from a full squad."
Glasgow Rugby backs coach Sean Lineen echoed Howells' sentiments, adding: "I think it is going to be pretty tight - there is a lot of hunger there as our sole goal is to secure a Heineken Cup place for next season.
"We have played the Warriors twice already this season - with one win and one loss - and when we went to their place we knew we had to get stuck into them, there is this real fear factor you are going to take a hammering when you go to Wales.
"There wasn't much free flowing rugby from either side on those occasions and, while we would like to win all our remaining games, we really have to target the two home fixtures.
"We will be the underdogs in all four games and we were desperately poor at home to Munster last time out and we must re-establish our previous home form immediately."
Glasgow Rugby have a four-point advantage over The Borders, who have the tough task of travelling to The Gnoll to tackle Neath-Swansea Ospreys.
"We desperately need a win - it is a huge game for us," said Tony Gilbert, The Borders Director of Rugby.
"It is vital we try and get ourselves back level pegging with Glasgow Rugby with the two Scottish Heineken Cup places on the line.
"We cannot do anything about the others but what we can do is try to change things and turn them around through our own efforts.
"There are four games left and every point - including losing bonus points - are going to be vital."