The Mail on Sunday says Boro will offer defenders Robert Huth and Chris Riggott to Newcastle in a bid to land Ameobi.
Boro manager Gareth Southgate has five centrehalves at The Riverside and hopes Newcastle's need for a defender will persuade them to let Ameobi, 26, go.
Germany's Huth was once a Newcastle target and is fit after ankle surgery which has restricted him to eight Premier League starts since a £6million move from Chelsea in 2006.
Riggott has been left out of the team in favour of Jonathan Woodgate and David Wheater this season.
Ameobi hasn't started a game since September and is behind Michael Owen and Obafemi Martins in the pecking order.