Kilgallon impressed with a polished display against Spurs in the Carling Cup last week. He was signed initially on a three-month loan deal in August by former Hammers manager Glenn Roeder.
It is due to expire in just over a fortnight's time, but Pardew will discuss the possibility of setting up a longer deal with Peter Reid before then.
"Matthew's very quick, he's got some nice attributes and he proved against Tottenham that centre-back is his best position," said Pardew.
"He's just lacking experience and I'd like to keep him to the end of the year, but that's up to Leeds."