Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
Rovers are looking for a new man to fill the Ewood Park hotseat after Steve Kean called time on his chequered 21-month reign as boss, stating that he was "forced" to step down and that his position had become "untenable".
Various names have been linked with Rovers, who are currently ninth in the npower Championship, over the last couple of days, with former striker Alan Shearer, ex-midfielder Tim Sherwood and much-travelled Swede Sven Goran Eriksson all linked.
Singh confirmed the club had been inundated with applications, telling Sky Sports News: "It has been a pleasant surprise despite the club having a lot of negative publicity.
"I would say it is an important job so I am pleasantly surprised there is a lot of people wanting to be a part of Blackburn Rovers.
"I would love to do it as soon as possible. It could be one week or three weeks. While we are in a hurry to appoint a new manager, we will not be rushed into it."
Despite the speculation, Singh refused to name any possible successors to Kean but suggested the list was full of quality.
"They are big names of course. Not so big names as well (are on the list), good coaches and it is a very interesting mix," he said.
"There are a lot of big names from football and definitely there are big, big names on that list. In this moment of time, I will not release any names.
"You can not ask a man of Sven Goran Eriksson's experience to apply for a job. That would be an insult to him. When we deem it fit to speak to someone who is in that category we will make the first approach."
In the meantime, Eric Black has the task of lifting Rovers up the table, with Singh fully behind the caretaker boss.
"Eric is now in charge of the team," he added.
"This is a very difficult time for Eric because he is still trying to instil his ideas on the team and it will take some time. Day-to-day management of the team is in the hands of Eric Black."