The midfielder, 36, who joined Boro for £5m in August 2007 and played over 100 games for the club, was part of a Bolton Wanderers side who dropped into League One last season during a grim campaign on and off the pitch at the Macron Stadium.
Boro's recent poor form has raised fears that they could suffer a similar fate to Bolton last term if they do not arrest their current slide, with another former Premier League regular in Stoke City also enduring strife at the wrong end of the Championship.
A wretched eight-game winless run, incorporating a pitiful return of just three goals, has seen pressure piled onto the shoulders of rookie Boro chief Woodgate, in his first job in senior management.
Despite recognising Boro's position, O'Neil feels that supporters should show patience regarding Woodgate's abilities to stop the club's autumnal malaise.
O'Neil said: "I think they need to give him (Woodgate) more time. They have obviously decided that he is the man for them and even though he is very young managerial-wise, he did have the experience of working under Tony Pulis last year.
"I just think that clubs need to be very careful with always changing. There has obviously been a huge cut in the budget so Jonathan Woodgate is not working with the same budget others had in previous years and with that has come a drop-off in performances and points return. I hope that Middlesbrough give him longer.
"It would be devastating if they were to go down and they clearly need to get themselves out of danger, but if he can do his bit and keep them safe, they need to then give him longer because he is doing well there with the younger lads.
"You cannot have a 14-week cycle of ripping things up and starting again because in another 14 weeks what do you do?
"There are two sides to this. Woodgate has to do his part and produce the best that he can with what he’s got while the club needs to give him time.
"Relegation though is unthinkable for Middlesbrough and you could never imagine them in League One so let’s hope he can turn it around."
Despite a testing time for Woodgate in what is turning out to be a baptism of fire in management, O'Neil is also confident that the 39-year-old's character and leadership qualities which he displayed in his playing days will stand him in good stead in his ability to transform Boro's season.
He added: "He was a leader in the dressing room and hugely respected. He was a fantastic player who has had a fantastic career, so you were drawn to him and respected that he’d played for Real Madrid.
"You could tell he was going to go on and manage: he was always studying things and talk intelligently about the games coming up. He always had a tactical view on them.
"So it is no surprise to me at all that he went down that road and he got a very big job as his first appointment. That can be tricky.
"Even Gareth Southgate who is now the England manager found it tricky when he got the Boro job straight after playing."
**Gary O’Neil, speaking exclusively to 888sport
Read the full 888sport interview with Gary O’Neil here.