The second legs are scheduled for April 14-15 which could mean Liverpool having to play on the day the city remembers the 96 people who died at Hillsborough.
UEFA denied media reports that it had declined Liverpool's request not to play on April 15.
"We are aware of the huge significance of the April 15 date for both Liverpool FC and their fans, and that is why we will do our utmost to make sure that the club does not have to play its UEFA Champions League second leg quarter-final match on that day," UEFA President Michel Platini said in a statement.
"I have been in regular correspondence with Liverpool FC on this matter for some time now and fully sympathise with the club's feelings as to the importance of respecting the date of April 15 for those who lost their lives at Hillsborough.
"This being the 20th anniversary of that tragic disaster in 1989 makes it even more relevant and we will take this into account."
Liverpool captain Steven Gerrard, whose cousin died in the crush before the club's FA Cup semi-final against Nottingham Forest in 1989, said on Sunday he hoped UEFA would show "common sense" over the matter.
"You would think that they would accept the significance of the date for the people of Liverpool but unfortunately feelings like that don't always come into it," Gerrard was quoted as saying on Liverpool's website.
The draw for the Champions League quarter-finals will be made on Friday.
