Forster is part of England's squad in Brazil, but the Celtic keeper has been linked with a move away from the Scottish champions before the start of next season.
Defender van Dijk, meanwhile, has also been the subject of speculation after an impressive first season with Celtic following his arrival from Groningen and has been linked with a transfer to England.
However, while Lawwell is unsure what will happen concerning Forster's future, he has stressed that Celtic do not want to sell van Dijk and he wants the centre-half to work with new manager Ronny Deila.
Lawwell is quoted in the Sunday Mail as saying: "We'd start by saying we don't want anyone to go, then if things develop we want to hold on to them as long as possible. Which would be to the end of the Champions League qualifiers.
"I wouldn't have thought anything will happen with Fraser until he is back from the World Cup.
"But with Virgil there has been nothing so that is good news. One year is far too early for him. He needs another year in the Champions League. He has to develop here.
"We have a manager whom I think will make him better, make all our players better, and if I was his advisor I'd be saying, 'No, no' - but you never know. And yes, I've spoken to his agent. We don't want to sell him.
"Unfortunately, though, in my experience you can say a player isn't going and then someone comes in the back door and he's offered four or five times his money.
"If nobody comes in then fantastic. If they do then we have a bigger problem."
Lawwell also confirmed that any contract decisions will be left in the hands of new boss Deila, with key man Kris Commons entering the final 12 months of his deal.
He added: "Kris has a year left so we're waiting to see what happens with the manager. The same applies to Lukasz Zaluska, Beram Kayal and Filip Twardzik.
"The rest are on two or three years so we are pretty stable. The only two risks are Virgil and Fraser."