"It did not take me too long to have a chat with them as everyone knows I have supported, played and managed the club," he told the club website.
"I think over the years a lot of people have linked me with the job a couple of times and there was a groundswell of hope.
"I know how things work in the club and now we've got to a stage where it is just a fantastic feeling to be back involved at the club again."
Despite the the lack of funds for his predecessor and the club running up a debt of around £30m, Jefferies believes he is taking over at "a great time".
"We're in the top half and we're in a cup semi-final. It is very positive," he said.
"It is not often a manager can come back and it has been nine years.
"Hopefully we can use the experience gained in that time. Although I am a bit older, the passion is still there for this club."
Laszlo steered Hearts to a third place finish in the Scottish Premier League last season but the Edinburgh side have struggled for form this term.
A good run of seven league games without defeat saw Hearts climb up the table to fifth spot but Wednesday saw them slump to a 3-0 home defeat to Aberdeen, who also knocked the Tynecastle side out of the Scottish Cup this month.
With just nine goals in 11 home league games this season, Laszlo had repeatedly spoken of his desire to add strikers to his squad in the January transfer window.
But, on the morning of his departure, he revealed that he would have no money to spend even if prize-asset Lee Wallace was sold.
"At the end of November or beginning of December, I gave to the board a complete plan for the short-term and also the long-term," he revealed.
Laszlo was frustrated by a lack of transfer funds
"Now the January transfer window is finished. This plan you can put it in the dustbin."
Laszlo also blamed the club's reliance on young players on the worst injury crisis of his career, which included a spell in charge of Uganda.
He said that being forced to persistently field youngsters resulted in them picking up injuries more physically developed players would have avoided.
"The body reaction showed I had it right," said former Ferencváros and Borussia Mönchengladbach boss Laszlo.
"This is not an accident that Craig Thomson, Johnny Stewart, (David) Templeton, Scott Robinson are injured."
That quartet is just the tip of the iceberg, with Andrew Driver, Lee Wallace, Ruben Palazuelos, Ian Black, Laryea Kingston, Arvydas Novikovas, Jamie Mole, Calum Elliot, Jamie MacDonald and Jamie Mole all nursing problems.
Hearts turned to Laszlo after then Motherwell boss Mark McGhee, now in charge at Aberdeen, rejected advances from Tynecastle.
Having trailed in eighth in the previous season, Laszlo immediately transformed an ailing side and clinched third spot by a margin of six points.
Romanov assumed control of the Edinburgh club in early 2005, with John Robertson, George Burley, Valdas Ivanauskas, Graham Rix and Stephen Frail the managers to serve him before Laszlo.
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