Dyke made the suggestion at a meeting with ITV executives last week.
"It was a cheeky suggestion to ITV because BBC football is doing so well, unlike ITV," a BBC spokesperson said.
"If they wanted to offload them we would consider it, but at the right price," she added.
ITV gambled that a peak-time Saturday football programme could win over viewers. It hired big-hitting presenter Des Lynam to front the show, which went out during peak viewing time on Saturdays at 7pm. The show had to be moved when it failed to draw more than an average audience of 4.4m viewers.
The show has now been replaced by Saturday evening ratings heavyweight Blind Date, which is fronted by Cilla Black and drew more than 7m viewers on the first show of the new series.
However, even in its new slot, at 10.30pm on Saturday, the Premiership football league highlights show only managed to attract 2.9m viewers, despite featuring high-profile games involving Manchester United, Liverpool and Arsenal.
Meanwhile, the BBC's live coverage of Sunday's FA Cup first-round tie between Hereford and second-division Wrexham, which kicked off at 1pm, drew 3.1m viewers.
ITV spent £61m a season for three years acquiring the rights to the highlights of Premier League football games. They were previously held by the BBC, which paid £40m a season to broadcast them on its long-running Match of the Day show.
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