Glenn believes that the left-sided midfielder, who wants to leave the French club after they were relegated to the second division and then banished to the third because of financial irregularities, could be a useful addition to the squad.
But he is also keen to stress that he working round the clock in chasing more established players.
"It's been a day spent in the office and I've lost track of how many calls I've made, following up enquiries and chasing possibilities about players we could bring into the club," he says.
"Some were pretty fruitless and others might lead to a deal being made, but we keep ploughing through it and I'm certainly looking forward to the time when we can conclude some more deals and get on with the training of the lads so I can get out in the fresh air rather than spend so much time in the office like I am at the moment.
"Fortunately Paul Goddard has been very impressive and taken to the job like a duck to water with Roger Cross and when I've been out there I've enjoyed working with the players - which is the best part of the job.
"I won't stop in my pursuit of buying players I feel we need in positions where I think we are weak and there are still two or three positions that badly need filling.
"I'm very determined and very hopeful that they will be filled by August 18th."
Talking of Lauren, he says: "He hasn't done himself any harm at all in the two games and daily training, and he plays in that wide left position where we haven't got too many players.
"I know for a fact there is another Premiership club waiting in the wings to sign him if we don't but having spoken to him he wants to stay with us because he has been impressed with the training and the club.
"If there is an opportunity to stay at West Ham he wants to take that up and I am hopeful of being able to do something so we've spoken to his agent and we will see what comes out of it.
"We've got first shout and if I can do something he will be a good addition to the squad - but we are also looking, of course, at some key signings in important areas of the team.
"If we could take Lauren in on a free though, that would certainly help us; at 22 he is the right age to bring someone into your club.
"Hopefully he will improve and play his part in due course."
Amidst rumours of a move for Morocco international Youssef Sefri he says: "Agents are using West Ham as a vehicle at the moment to tout their players around even if they know we are not interested because they are looking to see if there is anyone else out there willing to take their players on, and it is a nuisance."
Despite the attempt to sign Lauren, Glenn sees Joe Cole as first choice in that left midfield position, though he jokes: "I'd be happy with Joe Cole in any position other than in goal.
"But on a serious note I've been made responsible, if you want, for two of England's finest young talents and I'm looking forward to working with them on a regular basis.
"Michael Carrick is very settled in central midfield and one day Joe will also find himself playing in the central midfield.
"But at the moment I think he is more dangerous coming from a wide left position and he himself feels maybe that playing from wide left coming in on his right foot in a 4-4-2 is his best opportunity to get into the England team.
"You could say it is a position that is not exactly nailed down to any one particular player in the national team whereas probably the other three midfield positions are.
"He certainly did well against Mexico in that position."
Glenn says his trip to Sweden at the weekend was something of a wasted journey, adding: "The game in Stockholm was very poor but Stockholm was very nice!
"But you have to follow up every lead. We are in for a couple of players but don't ask me who because there will be no clues.
"We are pursuing a couple of possibilities and finding out from agents if players are interested in joining us."
And he reiterates that he will not pay over the odds for Hannu Tihinen, who he rates at around the £1.5m mark.
"Hannu is keen to join us and we are in talks with Viking Stavanger through his agent, but again the price is proving a little bit difficult," he says.