Newcastle wanted to loan Woodman out, as they looked to get the England youth international first-team experience.
However, Posh decided against signing Woodman, in favour of Reading goalkeeper Jonathan Bond.
“He was one of the options sent to us via an agent,” MacAnthony said.
“It came down to him and Bondy. We could have signed him. He’s a terrific, young goalkeeper.”
Woodman has not had a chance at Newcastle this season, with all his action coming for the Magpies’ development side.
Woodman has been unable to force his way up Rafael Benitez’s selection order, and ahead of either Karl Darlow or Rob Elliot.
Newcastle could now loan out Woodman in January, but whether Posh will still be an option for the youngster remains to be seen.
Bond’s current loan deal at Peterborough is due to expire in January, but MacAnthony has confirmed that he will attempt to negotiate an extension to that deal.
If Posh do manage to tie down Bond until the end of the season, Woodman will almost certainly have to look to another Football League club, if he is hoping for first-team experience.