Former Arsenal midfielder Vieira emerged as a target over the weekend, and the Magpies are expected to hold talks with him over the next 48 hours.
The French World Cup winner has been working with Manchester City’s development squad and Under-21s while he completes his coaching certificates, and was on a UEFA Pro Licence course in Wales last week.
His lack of experience would be a risk, but Newcastle owner Mike Ashley and managing director Lee Charnley are believed to be excited by the opportunity of getting a high-profile name in to take charge of the first team.
Whether the 38-year-old can be persuaded to move to St James’s Park is a different story, but he is not the only coach in the frame for the job, with Newcastle having drawn up a three-man shortlist for the job.
Also on the list is Steve McClaren, despite the former England boss having already turned down the job twice since Alan Pardew’s departure at the turn of the year.
McClaren initially rejected advances by Charnley in January, opting to stay with Derby County for the remainder of the season, and then turned down Newcastle again after the Championship ended when the Magpies wanted him to take control for the last three games of the Premier League campaign.
However, Charnley and chief scout Graham Carr remain fans of McClaren.
the 54-year-old, despite his dismissal by the Rams last week.
The third man in the frame remains unknown, but Charnley is understood to also be opening talks with him over the next 48 hours as he looks to get the appointment sorted before the end of the week.
Newcastle’s players return for pre-season training four weeks today - June 29 - which means time is of the essence in terms of getting their man in place.
While Carr and Charnley will be mainly responsible for signings, Newcastle want the new head coach to have some input into transfers, while pre-season fixtures also need finalising.
John Carver, who took charge for the last five months of the campaign which saw Newcastle plummet down the table, has already been in discussions with Charnley over pre-season. Carver is expected to stay at United in a coaching role, but is not being considered for the top job.