Neill, 34, has been on the lookout for a new home since he was released by United Arab Emirates club Al Wasl a fortnight ago.
A host of A-League sides has shown interest in signing the former Blackburn and West Ham defender.
However reports suggest Neill is in talks with the Heart and a deal could be done as soon as this week.
Heart coach John Aloisi is keen to shore up his defensive stocks after losing Curtis Good, Michael Marrone and Aziz Behich in the past year.
The Heart are sitting fifth and fighting to keep a spot in the top six ahead of Friday night's clash with Newcastle at Hunter Stadium.
Australian captain Neill has made 89 appearances for the Socceroos and featured at two World Cups.
He spent six years with Millwall in England before stints at Blackburn, West Ham, Everton and Turkish club Galatasaray before moving to the Middle East with Al Jazira in 2011.