The former England international is in charge of the Cherries while the club search for a replacement for Jason Tindall and has two victories from two games.
Last weekend, Woodgate brought to an end Bournemouth's four-game losing run with a 3-2 win over Birmingham City.
The South Coast side then claimed a Premier League scalp in the FA Cup fifth round on Tuesday night as they beat Burnley 2-0 away from home courtesy of goals from Sam Surridge and Junior Stanislas.
When asked whether his chances of taking the job full time were increasing, the 41-year-old stressed "let's not get carried away" and insisted that "football can come back and bite you". The former Real Madrid defender is currently third favourite with the bookmakers for the role, behind David Wagner and Patrick Vieira.
It will be a tough job to lead Bournemouth to automatic promotion this season as the Cherries, currently in sixth, are 10 points behind the top two, but they are five points clear of seventh-placed Middlesbrough.
The South Coast outfit are the Championship's second-highest scorers behind Brentford but have won just one of their last six league away games.
Bournemouth have also never won on their seven previous trips to the City Ground, the most recent of which was in 2015.
Forest have won just four times on home turf in the league in 2020-21 in what has been tricky campaign so far.
The Reds have picked up two crucial victories in the last fortnight away from home at Coventry City and Wycombe Wanderers to move up to 18th, four points clear of the relegation zone, but they have not won three successive league games since January 2020.
Chris Hughton's side picked up only one point from a possible 18 against the current top six in the first half of the season – losing 2-0 away to Bournemouth – and travel for another tough fixture at Swansea City next week.
Nottingham Forest Championship form: WWLDWW
Nottingham Forest form (all competitions): WLLDWW
Bournemouth Championship form: DLLLLW
Bournemouth form (all competitions): LWLLWW
Lewis Grabban is expected to be available again after missing the trip to Wycombe, but he may have to settle for a place on the bench given Murray's excellent full debut.
Jack Stacey is a doubt for Bournemouth after being forced off at half time against Burnley with a thigh problem.
Nottingham Forest possible starting lineup:
Samba; Christie, Figueiredo, Worrall, Bong; Cafu, Garner; Knockaert, Krovinovic, Lolley; Murray
Bournemouth possible starting lineup:
Begovic; Smith, Mepham, S. Cook, Kelly; L. Cook, Lerma, Wilshere; Brooks, Surridge, Danjuma
We say: Nottingham Forest 1-1 Bournemouth
Woodgate has halted Bournemouth's slump and will be hopeful of another win to aid his chances of getting the job on a long-term basis. However, Forest have proven hard to beat in recent weeks, and they are still in need of points to pull clear of relegation danger.