The deal for Jelavic is believed to be worth £6.5million with a further £1million due if the Tigers avoid relegation, eclipsing the club's previous biggest outlay of £5.25million for Tottenham's Tom Huddlestone last year.
And there could be more to follow with the Tigers also pursuing long-term transfer target Shane Long - although Press Association Sport understands West Brom have rejected the latest bid to lure him from The Hawthorns.
Despite his evident willingness to boost his club's Barclays Premier League survival bid, Allam chose Tuesday to once again reiterate his hard-line stance over his bid to change the club's name to Hull Tigers.
Allam warned he will be gone "within 24 hours" if his move does not receive widespread backing, telling Sky Sports News: "No-one on earth is allowed to question my business decisions.
"I won't allow it. I can give you my CV to give you comfort, for what I do in business, what I have achieved, but for someone to come and question me is not allowed.
"I'm here to save the club and manage the club for the benefit of the community. It will never, never be the other way round - that the community manage it for me.
"But if the community say go away, I promise to go away within 24 hours."
Allam added that the same applied should his move to have the club's name changed be rejected by the FA.
He said: "Still the same thing. It's a free country. No two ways about it. Have I ever said something and went back on it? No."
Allam has clashed with supporters over the plans and caused controversy in November when he reacted angrily to lobbying from campaign group City Till We Die by retorting: "They can die as soon as they want."
But Hull fans will have been placated by the imminent deal for Jelavic, who was allowed to leave Goodison Park by Toffees boss Roberto Martinez in order to pursue his World Cup ambitions with Croatia.
''He (Jelavic) will sign today - our biggest ever signing," Allam said. "The club is moving forward. We are paying a lot of money for first-class players.''
Jelavic had fallen out of the first-team picture following the summer loan signing of Romelu Lukaku but still proved he has an eye for goal when he scored two goals in the 4-0 FA Cup win over QPR earlier this month.
Jelavic made a stunning start to his Everton career after joining for £5.5million from Rangers in January 2012, scoring the winner on his full debut against Tottenham and going on to hit 11 goals in 16 games before the end of the season.
But he struggled to maintain his momentum during the following campaign and has found his first-team chances increasingly restricted.
Meanwhile, Hull's move for Long appeared harder to finalise with the Baggies clearly unwilling to let the Republic of Ireland international leave unless it is for the right price.
Long had been on the verge of joining the Tigers in September before a last-minute hitch in negotiations kept him at The Hawthorns, and the player reiterated this week that he is in no hurry to leave.
Long told the Birmingham Mail: "I have always said I'm happy and settled here and I've got family who like living in the area.
"I suppose if a club doesn't want you, what's the point in staying?
"It's business at the end of the day and we've got a lot of options up front. If they do want me to go we'll deal with that hurdle when it comes, or if it comes. But at the moment I'm a West Brom player and that's all I'm concentrating on."
- Sports & Recreation