Caretaker boss Mike Phelan expects the Hull board to make the right decision over the appointment of a full-time Tigers manager.
Phelan took temporary charge last month following the departure of Steve Bruce and, despite only having 13 senior players available, secured a second straight Premier League win at Swansea.
Newly-promoted Hull built on their impressive 2-1 victory over champions Leicester by winning 2-0 in south Wales, substitute Shaun Maloney and Abel Hernandez scoring in the final 12 minutes.
And Phelan concedes he can only fulfil his ambition of landing the job on a full-time basis by winning matches.
"I am sure the people in charge will do the right thing," said Phelan after Hull made it six points from six.
"I think everyone knows we have decisions to make on my future, the future of the staff and the ownership.
"Hopefully those decisions will be taken soon. It could be a busy time."
The travelling Hull fans were in no doubt that they want Phelan, the former Manchester United assistant manager, to succeed Bruce on a full-time basis.
As they secured their first top-flight away victory since winning at Crystal Palace in April 2015, the fans sang "we want Phelan in".
"It was nice to hear them singing that," said Phelan. "It's better than hearing them singing 'we want Phelan out'.
"I am sure if I do stay I will get the white handkerchiefs at some stage this season.
"But all I can do is win football matches and at this moment in time that's what we are doing.
"Credit to the players, they stuck to their tasks, got the result, and I'm delighted for everyone.
"You have to have belief in your players and your football club, although my experience tells me not to get too carried away.
"A lot has gone on but I haven't had to shield the players from anything. They have just done a great job."
Swansea manager Francesco Guidolin saw his side win their opening game at Burnley.
But after an encouraging opening half, in which they might have had two penalties, Swansea went down cul-de-sacs after the break and Hull goalkeeper Eldin Jakupovic had little to do.
"I am disappointed with that," said Guidolin.
"My dream was to win and I wanted to be on top of the table.
"The new owners were here - we had a meeting with them yesterday - and I am disappointed for them to see us lose in our first home game for them. Today was not a good day.
"But there are no excuses. It's a step back and we have to return to our work on the training field."
Swansea conceded regularly from set-pieces last season and were undone again in that department when Curtis Davies helped on Robert Snodgrass' corner and Maloney pounced from close range.
"We have to improve at set-pieces," said Guidolin.
"It's important we do that and I am confident in my players."
Source: PA
