Balogun - who was at the centre of controversy at the World Cup after US President Donald Trump successfully intervened to force FIFA to suspend a red card so he could play against Belgium in the last 16 - had undergone a medical in England before Everton started to raise some concerns about his fitness.
Although the Premier League side decided to proceed with the deal - after a couple of hours of discussions between the two clubs' medical departments - Balogun pulled the plug himself.
"Five minutes before the close of the (English) transfer window... I signed the contract, and Everton signed the contract," Monaco managing director Thiago Scuro told reporters.
"At the last moment, the player decided not to sign.
"And basically, the allegation of Balogun is that: 'I'm not feeling well by the way the club treated me. I'm not feeling well by the way the medical department of Everton brought doubts about my future or my health. I have nothing, so I'm not comfortable to sign a long contract here'."
Some reports had said that Everton had attempted to renegotiate the fee for the 25-year-old Balogun, but Scuro did not mention any such discussions.
Balogun had emerged as a late target for the Toffees after Guinea-Bissau international Beto completed a move to Serie A club Fiorentina.
His decision to back out has left Everton light on forwards for the new season, although they did secure the return of Jack Grealish on a season-long loan deal from Manchester City, for the second consecutive campaign. They also signed English defender Ainsley Maitland-Niles from French club Lyon.
Balogun caught the eye with his performances at the World Cup this year, but after a red card he received in the USA's defeat of Bosnia-Herzegovina in the round of 32 ruled him out of the last-16 clash with Belgium, Trump intervened.
The USA lost the game 4-1 and Balogun later admitted that the furore over the red card had affected the team's performance.
'Markets still open'
His decision to remain has left Monaco in a financial pickle, as they had budgeted for his sale.
They had discussed selling Senegal midfielder Lamine Camara to Chelsea if the Balogun deal fell through, but had pulled out of that when they thought their striker would leave - which he may yet do as some leagues have yet to shut their transfer windows.
"We still have other markets open, and we are having discussions about different players, including Balo," said Scuro.
"Let's see what can still happen until Thursday because we still have markets that have the financial capacity to do something."