Steven Lawless has been given a new deal by Partick, a few days after being handed a ban for breaching betting rules.
Midfielder Miguel Pallardo has signed a new one-year deal with newly-promoted Hearts.
McDermott is looking to sign an established Barclays Premier League player to assist the club's battle against relegation, but denied Welbeck is on his wish list.
The 22-year-old is behind Robin van Persie, Wayne Rooney and Javier Hernandez in the Old Trafford pecking order, leading to speculation he would be loaned to another club until the end of the season.
McDermott distanced himself from one of the more unlikely transfer stories of the January window.
"I can't see Sir Alex Ferguson sending Danny out to any club. I don't know where this has came from, it's got nothing to do with me," McDermott said.
"Welbeck's a great player and I'm sure Sir Alex Ferguson thinks he's a great player too. Everyone in the country thinks that."
"We're still looking for an experienced player and if we can find one who has played some games in the division that would be good for us," he said.
"It's never easy to find a player like that, but it's definitely on the agenda."
The arrival of former Portugal Under-21 international Carrico, who had fallen out of favour at Sporting Lisbon, and Hope Akpan from npower League One club Crawley has been greeted with a lukewarm response from supporters.
Carrico, 24, is set to make his debut against West Brom on Saturday and Akpan, 21, could also figure with McDermott insisting they are already of Premier League standard.
"The fans were probably critical when we signed two players from Cork many years ago - Kevin Doyle and Shane Long," said McDermott, who is continuing to track Central Coast Mariners attacking midfielder Tomas Rogic and Swansea striker Danny Graham.
"I believe that if the player's got qualities and can play in the Premier League then whatever division they come from it doesn't matter.
"Daniel Carrico looks ready for a start and he says he is. He's a good player, can pass well, reads danger and is a leader.
"We feel that in Hope Akpan we've got another young lad who can do really well for us."
Former Reading striker Long will be returning to the club he left in 2011 when West Brom visit the Madejski Stadium.
Long, who was sold for £6.5million after the Royals had failed to secure promotion to the Premier League, has proved a hit at the Hawthorns, scoring nine goals in 24 appearances this season.
"I speak to Shane regularly, he's a friend. The impact he's had since 18 years of age has been phenomenal," McDermott said.
"Sir Alex Ferguson mentioned him the other day and talked about what a top player he was and he knows what he's on about.
"He might end up at a top-four club, that wouldn't be beyond the realms of possibility.
"I remember when I first put him in the team he wasn't scoring but he was giving us so much. But then he started to score as well and he was phenomenal for us.
"He's exactly what we're about. You can say it in two words: Shane Long. I wanted him to stay but we couldn't afford to keep him. We had to sell to survive."