"The January transfer window is feeding time for the sharks," Gibson said today. "But we will fight to keep this squad together because we know we have a lot of quality here.
"I believe that Stewart Downing and in fact all of our players should stay and fight to turn this club around.
"There is a enough talent here to lift the club back up the table and we want to give it every chance to shine through.
"The players owe it to the club and to the fans and they owe it to the manager."
Tottenham are understood to have bid £11 million for Downing but Gibson has warned Tottenham chairman Daniel Levy the 24-year-old winger is not for sale.
"As far as Stewart is concerned we understand the background that has led to this transfer request and we believe that his head has been turned," Gibson continued.
"We will work hard over the next few weeks to try to convince him to stay.
"I am adamant that nobody will be leaving in January and that Gareth Southgate will be working with the same squad of players at the beginning of February."
Meanwhile, former Boro striker Bernie Slaven has backed Downing's decision to submit a transfer request because he claimed their prized asset did not have a "hope in hell's chance" of fulfilling himself at the Riverside.
The former Republic of Ireland international spoke out after Downing's father, Stewart snr, claimed his son wanted to leave his hometown club less than 12 months into a five-year contract, due to frustration at the club's transfer policy.
The 24-year-old, who has played in 164 games for Boro since breaking into the first team in 2001, submitted the transfer request after Spurs lodged an £11 million bid for the international that was rejected by their relegation rivals.
His contentious move would not have gone down well with most Boro supporters but Slaven has supported the club's most high-profile player.
"As an ex-player, if I was in Stewy's position I would probably have done exactly the same," said Slaven, who made 307 appearances for the Teesside club between 1985 and 1993.
"My love for the club would be there – as Downing's is – he wears the shirt with pride.
"When he signed his five-year contract last year, hand on heart I didn't believe he would see the five years out. I thought we might get a year, possibly two.
"But we've seen the departures of Mark Schwarzer, Lee Cattermole, George Boateng and Luke Young since the summer and that has not helped Stewart's mindset."
Slaven, who nowadays makes a living as a radio presenter in the North-East, added: "He's 24, he's ambitious, he wants to play in Europe and he wants to get as many England caps as possible.
"Can he achieve his goals at Middlesbrough at the moment? I'm being brutally honest, and for the foreseeable future, there isn't a hope in hell's chance."
Downing is due to be in action for Boro this Saturday when they host Sunderland in what is bound to be a nerve-jangling contest between two sides near the foot of the Premier League table.
Gary Gil, the former Boro defender, is confident that Downing will not allow the current controversy to hamper his contribution to the Boro cause and urged fans to rally behind him.
"I just think it is important that the fans even if they are not happy with what has gone on this week think of the bigger picture," Gill said.
"They must think about performance on the day, getting the points against Sunderland and that means getting behind all your players."