“Everyone wants to calmly resolve the situation,” Davie said in a BBC interview.
Lineker was forced to “step back” from his duties presenting the flagship Premier League highlights show after accusing the UK government of using Nazi-era rhetoric in tackling illegal immigration.
The BBC said on Friday that England’s fourth highest goalscorer of all-time had breached guidelines on impartiality and the corporation would seek “an agreed and clear position on his use of social media” before an on-screen return.
However, the decision caused chaos to scheduled sports programming across the BBC’s television and radio output.
Former England strikers Ian Wright and Alan Shearer were among the pundits who refused to take up their usual roles on Match of the Day, followed by the programme’s commentators.
As a result, the longest-running football television programme in the world was aired for the first time without a presenter, pundits or even commentary in a shortened 20-minute highlights package of six matches from the English top-flight.
Weekend preview show Football Focus and results programme Final Score were also pulled from the schedule, while BBC Radio 5Live’s coverage was disrupted.
When asked if he should resign over the crisis, Davie replied: “Absolutely not.”
“I think that my job is to serve licence-fee payers and deliver a BBC that is really focused on world-class impartial landmark output, and I look forward to us resolving this situation and looking forward to delivering that.
He added: “To be clear, success for me is Gary gets back on air and together we are giving to the audiences that world class sports coverage which, as I say, I’m sorry we haven’t been able to deliver today.”
British Prime Minister Rishi Sunak said he hoped the stand-off can be “resolved in a timely manner.”
Lineker is a freelance broadcaster for the BBC, not a permanent member of staff, and is not responsible for news or political content so does not need to adhere to the same strict rules on impartiality.
The former Leicester striker was in attendance at the King Power Stadium to watch his home town club lose 3-1 to Chelsea, but did not speak to reporters.
Some Leicester fans showed their support for Lineker with placards reading: “I’m with Gary, migrants welcome.”
The row was sparked by Lineker’s response to a video in which Home Secretary Suella Braverman unveiled plans to stop migrants crossing the Channel on small boats.
Lineker, the BBC’s highest-paid star, wrote on Twitter: “This is just an immeasurably cruel policy directed at the most vulnerable people in language that is not dissimilar to that used by Germany in the ’30s.”
The Conservative government intends to outlaw asylum claims by all illegal arrivals and transfer them to other countries, such as Rwanda, in a bid to stop the crossings, which totalled more than 45,000 last year.
A YouGov poll published on Monday showed 50 percent backing the measures, with 36 percent opposed.
But rights groups and the United Nations said the legislation would make Britain an international outlaw under European and UN conventions on asylum.
The BBC’s move sparked a wave of criticism from politicians and public figures, many of whom accused it of buckling to demands from Conservative lawmakers.
Opposition Labour leader Keir Starmer said the BBC “got this one badly wrong and now they’re very, very exposed”, while a petition calling for Lineker to be reinstated has attracted over 190,000 signatures.
The issue has brought to a head years of debate over BBC impartiality, which intensified after Britain voted to leave the European Union in 2016.
The Lineker row comes at a particularly heated period after allegations that BBC chairman Richard Sharp facilitated a loan guarantee for former prime minister Boris Johnson while applying for the job.