A loan fee of £2.7m has been agreed for the 23-year-old, according to Sky in Italy, but Roma want Leicester to commit to buying him for £22m in a year's time.
Sky Sports News has been told there are still a variety of options being considered, on a sliding scale of commitment from the buying club's perspective.
Leicester's priority remains a centre-back - a position where the squad is particularly bare - and so the hierarchy is reluctant to spend big money up front for Under, or any other winger.
An 'obligation' to buy means Leicester would be forced to pay out when Under's season-long loan came to an end.
However, the alternative, which would seem to be more attractive to Leicester's bosses, would give them the 'option' to buy - a free choice for the club to decide whether they want to purchase the winger on a substantive deal.
A compromise seems the most likely outcome, whereby Under's move would only be made permanent if he plays a certain number of games for Leicester while on loan.
That scenario is, however, likely to mean a higher overall fee that Leicester would have to pay.
Under is a left-footed winger who has primarily operated on the right wing for Roma since arriving from Istanbul Basaksehir in 2017.
He has scored 17 goals in 88 games for the Serie A club, who are willing to allow him to leave as he has failed to hold down a first-team place during his three years in the capital.
In addition, Leicester have not given up on signing Burnley centre-back James Tarkowski.
Sky Sports News has been told that the two clubs have been in regular contact but remain apart on their valuations of the 27-year-old.
Brendan Rodgers wants to sign a new centre-back before the transfer window closes on October 5 and has been interested in signing Tarkowski since Harry Maguire moved to Manchester United last summer.
West Ham are also keen to sign Tarkowski and have had two offers turned down by Burnley.
Sky Sports News reporter Rob Dorsett...
Leicester may wait until later in the transfer window, in the hope that prices fall, as they look to strengthen their squad for an assault on four competitions.
A lot of Foxes fans are questioning why the finances are so tight, after the club received £130m from the sale of two key players - Harry Maguire to Manchester United and Ben Chilwell to Chelsea - in the last two summer windows.
There is not one simple answer to that question.
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The summer transfer window will run for 10 weeks from July 27 and close at 11pm on October 5.
A domestic-only window for trades between the Premier League and EFL then runs from October 5 and closes on Friday, October 16 at 5pm. Follow all the news and analysis on Sky Sports News and across Sky Sports' digital platforms, including with our dedicated Transfer Centre blog.