The 19 year old, who at 6ft 2ins will add further height to the Hammers rearguard, made his Premiership debut against West Ham at Elland Road in February and also played against Aston Villa last season.
Says Glenn Roeder: "As well as trying to sign some permanent players on our terms, we know we are going to make some short term loan signings.
"The squad is very small at the moment and we have been looking at the possibility of signing a highly promising young central defender for a while.
"We have had our eye on Matthew for a number of games and have watched a number of Leeds' pre-season matches where he has figured for their first team.
"He has played against Bristol City and Hull City and we finally decided to make a move for him this week. He had an outstanding game against Jason Roberts, the West Bromwich Albion striker recently.
"We are confident that if we use Matthew in our first team he will be up to the job and Peter Reid, who rates him highly, was happy for him to come to us for three months.
"He wasn't happy for him to go on loan for the whole year in the manner that we have been able to secure Neil Mellor, though we would have liked him for the full season.
"But he is a Leeds player and Peter has the call - he wanted to make it no longer than three months and then we can reassess things then.
"Peter is hoping he can gain experience the way Glen Johnson did at Millwall last season.
"He is a left-footed player and we have been looking at bringing in one for some time to the defence; we beat off competition from several clubs, and we need cover for Ian Pearce, Tomas Repka, and Christian Dailly.
"We are sure that when he is called upon he will be able to play first division football."
Ironically, Peter Reid has just beaten Glenn to the signature of Jermaine Pennant on loan from Arsenal, and Glenn admits: "It is wrong to speak about other people's players but I will say briefly that he is a young player with a big future in the game that we know is not for sale - but we would have liked him in our squad to gain valuable experience.
"But, because Premiership clubs can loan to each other now, it has not helped, and if a young man can go to another Premiership side to gain a first team place he will do that first.
"Carlton Cole has just gone to Charlton from Chelsea for a year and had that rule not changed I am sure he would have gone to a first division club for a year.
"He and Jermaine are two significant loan deals that involve highly talented youngsters and with the window coming up these things are starting to happen."
Glenn is looking to bring in another player this week to replace Rufus Brevett, who will be out for a number of months with an ankle injury.
"As soon as I realised the scan revealed a fracture I was contacting several clubs for a replacement - for at least a long term loan if not a permanent transfer," says Glenn.
"More than likely it will come through the loan system and we are working extremely hard to get a left-back in by Saturday."
Regarding Rufus' injury, Glenn reveals: "We have to be patient. It will take 10-12 weeks for the fracture to heal completely and as it is on a weight bearing bone we don't want repercussions.
"But it will mean that he can't put his body weight on it and, although you can't be specific, it will be four or five months before he plays again.
"He will be in plaster for four weeks and there will be a fair bit of muscle wastage on the calf.
"He will then have a lightweight cast for four weeks and then hopefully be able to start his rehab and build his strength up.
"It is not an injury you can rush and we need to give him the full complement of time to heal.
"But we hope he will be back before the new year and the most important thing for us and Rufus is that we have been told categorically that it is not career threatening."
West Ham have been mentioned in connection with West Bromwich Albion's Welsh international midfielder Andy Johnson this week, but Glenn says: "There has been a lot of speculation and we know the media realise we are looking for players so we are getting linked with a fair amount of them."