Lagrell told Reuters he would like the officials to be able to watch the television pictures viewers see at home but ruled out introducing goalline technology and video replays.
He has presented the idea to Sweden's top-flight clubs and if they agree, the FA would then need to get approval from world football's governing body FIFA to implement the plan.
"We don't think that we will ever have them (goalline cameras) in football as it means too many breaks in play, studying film and so on," Lagrell said in an interview at the Swedish FA's headquarters.
"But we have seen another possibility that we can ask to be a trial country for, and that is to use the regular TV signal.
"We think that it (goalline technology) is a worse alternative than simply giving the fourth official the possibility to look at the TV pictures.
"With that (use of the live television pictures), you have a team of officials who referee the game exactly as it is today, but this way they get something to help them."
Swedish referee Martin Hansson was widely criticised for missing Thierry Henry's handball in the lead-up to a French goal that denied Ireland a place at this year's World Cup.
Lagrell said this could have been avoided by watching the TV pictures.
"That particular problem could have been solved using this system," he said.
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More high-profile misses followed during the World Cup in South Africa, notably England midfielder Frank Lampard's goal against Germany that was well over the line but was not given.
The mistakes led FIFA president Sepp Blatter, for years a stubborn opponent of goalline technology, to say the file on the subject would need to be reopened.
Lagrell said the size of FIFA and the differing levels of technological advancement in its member countries meant reaching a consensus would be difficult.
"It's no coincidence that we're asking about this because we are one of the countries that have TV production crews at every match in the Allsvenskan (Sweden's top division)," he said.
"It wouldn't be possible in the Superettan (Sweden's second tier) as some games don't have TV cameras present.
He added that comparing football to sports like rugby, which has used technology for some time, did not help.
Lagrell is awaiting feedback on the idea from the clubs before he can go any further.
"If the clubs in question think that it's a good idea, then we're prepared to ask if we can have a trial period in this country for one or two years," he said.
"And if we're not allowed, we have made an effort to fix it (the mistakes) and that's that."
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