Munster are being investigated over their management of scrum-half Murray's head injury in the Scotstoun game.
Ireland's Murray was allowed to return to the field after belatedly being sent for a head injury assessment.
"I was fully conscious and talking to them (Munster's medics) about my neck," said the Lions half-back on Wednesday.
Munster's 14-12 victory guaranteed the Irish province a place in the Champions Cup quarter-finals.
Recently, Northampton were found to have breached rules by allowing Wales winger George North to play on after a head injury.
Sale are also being investigated for an incident involving TJ Ioane.
On Sunday, European Cup organisers confirmed to BBC Sport that an Untoward Incident Review Group would convene this week to decide whether Munster had broken concussion management protocols.
However, Murray told a news conference he has been closely monitored since last weekend and has continued to pass all head injury assessment protocols.
"They (the Munster medics) didn't suspect that I had lost consciousness, which is the most important thing, and I came back on and finished the game and have felt fine since.
"I've passed all the tests and I'm fine. We've followed all the HIA protocols and worked with World Rugby.
"There's been a bit made about it but I am fine, I've trained fully today and the physios have looked after me, kept a close eye on me and I came through ok, so I'm ready for the weekend."
Murray emphasised that he is, like all players in the professional era, "very wary of concussion and how dangerous it can be".
"If I had a headache or I felt a little bit slowed or any of those symptoms, I'm fully sure I'd bring it up and I'd tell the physio and doctor and we'd deal with it and probably wouldn't play this weekend because everyone's aware of how dangerous concussion can be down the line."
Conor Murray has insisted he did not lose consciousness in Munster's European Champions Cup victory over Glasgow last weekend.