The club's manager held another round of talks with managing director Lee Charnley yesterday after Sunday's face-to-face meeting with owner Mike Ashley.
And the latest discussions, which come in the wake of emotional calls from supporters for him to stay, have been described as "positive".
Benitez – who has a break clause in his contract allowing him to leave this summer – is keen to see the club restructured, and that could spell the end for chief scout Graham Carr.
Charnley and Benitez also plan to discuss potential Championship signings and budgets this week.
However, interest in Benitez, linked with the vacant Everton post, from a Premier League club could yet scupper the club's attempts to persuade him to stay on Tyneside.
The 56-year-old said at the weekend that he must make a decision using his head as well as his "heart".