Parker, 26, lost his WBO world title to Joshua and is now ranked fifth by the organisation (as well as seventh by the IBF and sixth by the WBC). Russian Alexander Povetkin is the current No1 contender with the WBO and is seeking the big pay day of a fight against Joshua.
Povetkin fought on the undercard of the Joshua v Parker fight at the Principality Stadium and while he won with a dramatic knockout which knocked opponent David Price senseless, Parker and his team would have seen enough to know the Russian is not on the same level as Joshua.
Jennings, 33, has lost only twice as a professional – a points loss in 2015 to Wladimir Klitschko, who invited Parker to spar with him at his Florida camp beforehand, and a TKO against Luis Ortiz eight months later.
Jennings doesn't have a great knockout record – 13 of his 23 wins have come inside the distance – but he is considered to be very good defensively. He has good movement and footwork and would present a very difficult challenge to Parker, who would be under pressure to win and win well.
The other option for Parker is Alexander Ustinov, the giant Russian who ended David Tua's career in 2013. Should that be the case the fight would almost certainly be held in New Zealand.
Parker is likely to fight twice this year and it would be extremely unlikely that both of those fights would be in New Zealand.
"Whilst we are in negotiations with the Jennings camp, we have other options on the table, including Alexander Ustinov," Duco's David Higgins said.
"No deal will be made on the next fight until Joseph and his team are satisfied with the final terms and a contract has been fully negotiated and signed."