"We've dealt with the situation, Gary is staying... and that's all I'm worried about," said Blackwell.
"At this moment in time I would be loath to lose a senior member of staff so close to the beginning of the season."
Speed, 40, was interviewed for the Swans job last summer but eventually missed out to Paulo Sousa.
But he has been in the frame again after Sousa quit the Liberty Stadium last week to take charge of Championship rivals Leicester City.
It appears Swansea made Speed their first-choice target having coming in for a second bid which was refused on Monday.
Eighty-five cap Speed, who was born in Mancot, Flintshire, has gained his Uefa pro licence coaching qualification and has made no secret of his desire to one day become a manager.
As a player, Speed featured for Leeds United, Everton, Newcastle United and Bolton Wanderers before joining the Blades, becoming the first footballer to reach 500 Premier League appearances.
A source close to Speed said on Sunday that he was "disappointed" to be refused permission to talk to the Swans, but Blackwell insists he is happy at Bramall Lane.
"Gary is a level-headed lad, he wants to learn, and he can, and he's thoroughly enjoying his time here," Blackwell told BBC Radio Sheffield.
"We believe that Gary will be staying with us for next season and we're delighted with that.
"Everything moves forward. We've had to rebuild the whole backroom staff throughout the summer.
"We've had physios and fitness coaches headhunted by other clubs and we've just put all that back in and I'm just loath to losing anyone else."
Former Reading and Watford manager Brendan Rodgers has been interviewed for the Swans job, while Scunthorpe United's Nigel Adkins has been heavily linked, but Hamilton boss Billy Reid has turned down the chance to talk to Swansea.
Source: BBC Sport
Source: BBC Sport