Former Rangers midfielder Davis has played under the man linked with the vacant Ibrox post for the past six years.
O'Neill's future is uncertain after the Green and White Army failed to qualify for the 2018 World Cup this week.
The 48-year-old is under contract until 2020, although according to Sky Sports, the IFA are ready to offer him an improved deal to keep him out of the clutches of others.
But Davis, who has played under the Portadown native since 2011, hopes he will resist any offers that come his way.
"I speak for everyone by saying we’d love him to stay and how important he is for us," said the Southampton midfielder, as quoted by the Scottish Sun. "He has a bit of time and decisions to make.
"Other nations and clubs will probably take an interest in him because of the success we have had, but from a selfish point of view we really want to keep him."
David added: "As a manager, he’s unbelievable, we all love him and every single player plays for him. He’s such a good manager and it would be a shame to lose him.
"But we know on the other side, management is such a difficult job and if something came up, where it would be difficult to turn down, no-one would ever begrudge him that."
Davis, who won his 100th Northern Ireland cap last week, spent four years on Rangers books, but returned to English football after the club's demotion to the fourth tier.
And the 32-year-old would surely to back O'Neill's move to Ibrox, should he opt to leave his current role.
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