In an interview with online outlet Tag24, Lotito said that Milinkovic-Savic begged him for "a change of scenery" after eight years in Rome which would reportedly involve him earning 60 million euros ($66 million) over three seasons.
Italian media report that Al-Hilal have agreed to pay 40 million euros for the Serbia midfielder, 28, whose contract with Lazio expires next summer.
Lotito said Lazio had received an offer which was "a little bit less than 50 million euros".
"When someone leaves to go to Saudi Arabia their motivation is money, but you need to ask him why he wants to leave," said Lotito.
"I tried my best to keep him here but when he asks to go for his own personal reasons what can I do? We were willing to extend his deal and increase his salary.
"At the end of the day it's the player who decides what to do with his life. He phoned me begging me to let him go."
Fan favourite Milinkovic-Savic has scored 69 goals in 341 appearances since joining Lazio for 10 million euros in 2015, winning the 2019 Italian Cup and two Italian Super Cups.
He was one of Lazio's stars as Maurizio Sarri's team finished second in Serie A and qualified for next season's Champions League.
The Saudi Pro League has begun to snatch big names from the major European leagues thanks to the backing of the country's sovereign wealth fund.
Should Milinkovic-Savic make the switch to Al-Hilal he would join a number of other familiar faces enticed by the prospect of huge salaries.
Last week Brazilian forward Roberto Firmino became the latest to move to the Gulf state after signing for Al-Ahli, while former Liverpool captain Steven Gerrard took over as coach at Al-Ettifaq.
Karim Benzema and N'Golo Kante have also opted for Saudi clubs since Cristiano Ronaldo's move to Al-Nassr in January started the trend.
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