The Allams have pledged to take over the club's debts, believed to be around £20million, and make funds available to manager Nigel Pearson in the January transfer window.
The club themselves have yet to confirm the deal but a statement from the Allams read: "Businessmen Assem and Ehab Allam have agreed a Heads of Terms arrangement with Hull City owner Russell Bartlett to take a controlling interest in the Championship club.
"The deal was reached at 7pm last night (Tuesday) after several weeks of negotiations, leaving only the legal processes to be completed by the end of November."
Allam Sr later told BBC Radio Humberside that he expects the legal process to take up to three weeks.
He said: "It's been passed on to the lawyers and the financial people to start the due diligence and the legal procedure.
"You need two to three weeks with the legal and financial procedure and Christmas is five weeks from now - if we missed the opportunity then we would have lost the lot. Both sides had to make compromises knowing time was not in our favour."
He pledged to take a responsible approach to his prospective stewardship, saying: "My role is to take responsibility for the debts and pay them off, thus removing the burden so the club only has one thing to worry about - performance.
"Your expenditure should be within your income - no more borrowing after that. You want more money? Get more advertising, get promoted to the Premier League - but no borrowing."
On that promotion aim, though, he stressed: "I'm not interested in rushing it, I don't like the idea of up and down, up and down. You need another season to put yourself on very strong ground so you can go to the Premier League and stay there."
City have won just three of their first 16 league games under Pearson's stewardship, leading to speculation in some quarters over his future, but Allam added: "I think he's safe, but this is a decision for other people."
Allam left his native Egypt in 1968 after speaking out against dictator Gamal Abdel Nasser's regime and studied economics at the University Of Hull before working for Tempest Diesel as an accountant.
He worked for local company Tempest Diesel before buying it out in 1981 and renaming it Allam Marine. The company's turnover reached £116million last year and Allam's fortune is estimated at £150m.
On his motivations for the takeover, he said: "Forty-two years ago I came here, I was hurt by a military regime and tortured.
"I was welcomed, my family were welcomed, all three of my children had a university education in Hull. Why not pay back?
"I want to have the capacity increased to 40,000. Increased capacity will help - gradually - to reduce the value of tickets. I hope I will live to see football watched by every housewife and child - make the tickets £10, £5.
"Watching football should be like breathing air - free of charge."
The deal will see Bartlett and head of football operations Adam Pearson remain at the club as non-executive directors alongside the Allams.