The Republic of Ireland international only moved to Upton Park in a £285,000 deal from Wimbledon 12 months ago and The Hammers are now set to make a profit on the forward.
The two Championship rivals have concurred on a £500,000 fee for the 27-year-old, who has been targeted by Foxes boss Micky Adams to bolster his strike-force.
Adams has lost Paul Dickov, Les Ferdinand and loanee Marcus Bent since The Foxes' relegation from The Premiership last term.
But he is now due to receive a welcome boost with the acquisition of Connolly, who is due at The Walkers Stadium for a medical and to discuss personal terms this week.
"This move for a player like this shows that those who may have written us off in the transfer stakes should think again," Adams told the club's official website.
Leicester's swoop for the attacker will disappoint Sunderland, who had been keen on Connolly but they are unable to match The Foxes' spending power.
Adams is also continuing his efforts to add depth to his squad by taking a look at former Derby County defender Youl Mawene.
The Frenchman recently left Derby and will feature for Leicester on their pre-season tour to Finland.
He is expected to play against FC Espoo on Tuesday evening and links up with another defensive trialist - Chris Makin.
