The Warriors, however, have made a good start to the season and a triumph of their own could see them rise into the play-off positions.
Also in Scotland on Friday, Edinburgh host Heineken Cup holders Leinster, who recently have not fared well on Scottish soil.
The hosts, now coached by former Connacht boss Michael Bradley, will be hoping to build on a record that has seen them lose just once in their last six games at Murrayfield.
In the final game on Friday night, Aironi travel to Musgrave Park still looking for their first away victory - the home of the reigning league champions, Munster, may be a difficult place to find it.
The remaining three games take place on Saturday. Ulster, another team looking for their 100th victory in the competition, will have to overcome Scarlets in Llanelli to achieve that mark, never an easy task.
The Newport Gwent Dragons travel to Italy to face Benetton Treviso, and will be hoping to repeat the success of their last trip to the country in February when they defeated Aironi. Treviso, however, are enjoying a successful streak having won their last three games.
The round wraps up in Galway with Connacht hosting the Cardiff Blues. A home win, depending on other results, could see the teams swap places and Eric Elwood's men returning to the play-off zone.
Friday, October 28 -
Glasgow Warriors v Ospreys, Firhill Stadium, 7.05pm (live BBC 2 Wales)
Edinburgh v LEINSTER, Murrayfield, 7.35pm (live BBC ALBA)
MUNSTER v Aironi Rugby, Musgrave Park, 8.30pm (live TG4)
Saturday, October 29 -
Benetton Treviso v Newport Gwent Dragons, Stadio di Monigo, 3pm
Scarlets v ULSTER, Parc y Scarlets, 3pm
CONNACHT v Cardiff Blues, the Sportsground, 6pm (live RTÉ Two/S4C)