The striker has scored 20 goals and is currently the Ligue 1 topscorer. His goals have helped propel Montpellier to the summit and they currently hold a five-point lead over Paris St Germain with four games remaining.
Liverpool are keen to bolster their strikeforce as star striker Luis Suarez has only scored 11 goals in the Premier League this season. Giroud would likely partner Suarez as the Reds will aim to get back into contention for the Champions League in the coming season.
The Daily Mail reported that Kenny Dalglish and assistant Steve Clarke made the visit to Toulouse in order to see the striker in action. The report states that the French striker could be available for as low as £13.5 million.
However, Giroud has also attracted interest from Arsenal and Newcastle in recent months. Arsene Wenger has reportedly instructed his France-based scout Giles Grimandi to keep a tab on the player's progress.
And Montpellier owner Louis Nicollin dismissed Newcastle's attempts to court the services of his star striker last year declaring the northern club to be "too small" for Giroud.
The only stumbling block for Liverpool could be the fact that they will be unable to offer Champions League football for the striker next season. Should Montpellier win the Ligue 1 title, he will be able to participate in the competition with them while with Liverpool, he will have had to settle for the Europa League as a dismal season for the Reds has seen them struggle to stay in the top half of the table.
Liverpool are also said to be interested in Toulouse's Moussa Sissoko and Dalglish made the visit in order to evaluate the midfielder for himself as well. Sissoko is a tall and confident midfielder capable of making purposeful runs into the box and he could serve as a good deputy for captain Steven Gerrard.
Sissoko admitted after the game that Liverpool have made contact with his agent but for now, he insists that he wishes to stay with the French club.
"I had contact with English clubs, notably Liverpool, but I preferred to commit myself to Toulouse, the club where I have been educated," he told reporters after the game.
"It is up to me to go on improving, to become a world-class player."