And he will be urging his Warriors side to "do something special" at the Liberty Stadium tonight when they take on the Ospreys in the first of the Magners League Play-Off semi-finals.
"We are the poor cousins of Celtic Rugby at the moment and we want to change that. We have to give a good account of ourselves tonight," said Lineen.
"We want to do something special not just for Glasagow Warriors, but for Scottish rugby as well. We know we have to lift our game on all sorts of levels because the Ospreys are an outstanding outfit.
"We have huge respect for them and we know they are incredibly physical. We will need a strong hand from the referee and his touch judges.
"We mustn't get intimidated and we must get the basics right. They will have home advantage, but our form on the road this season isn't bad with four wins, two draws and three defeats."
Warriors skipper Ali Kellock felt his side might have snatched all the points the last time they played in Swansea, but he knows that after an indifferent start to their season the Welsh region have really picked up steam.
The Ospreys lost their opening two games against Ulster and Leinster, beat Edinburgh and then drew with the Warriors on October 30. But since then Munster, Cardiff Blues, Connacht, the Scarlets and the Dragons have all been beaten at the Liberty.
They also won all three home games in their Heineken Cup Pool, beating ASM Clermont Auvergne, Viadana and Leicester Tigers.
"We were very unlucky not to get a win the last time we went to the Liberty Stadium when it ended as a 9-9 draw. But the Ospreys' home form speaks for itself and we know we are going to have to be at our very best to get anything out of the game," admitted Kellock.
"We have been building quite successfully over the past few years although we had a bit of a blip last year. We are pleased to have got into the top four, but we don't want to sit back - we are here to try to win some silverware."
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