An intial bid to have the sanctions lifted was rejected by the Scottish Professional Football League (SPFL) at the end of 2013, with the game's governing body insisting that the club must serve the full term of the punishment for entering administration in the summer.
However, Jackson will speak to the SPFL secretary Iain Blair on Monday with a view to setting up a formal meeting, in the hope that "the change in circumstances" will prompt a different verdict.
BDO are hopeful that they can put forward a more compelling case for the restrictions to be lifted at the second time of asking, allowing Hearts to add some much-needed depth to their paper-thin squad.
It is understood the argument will be based on the fact there is minimal danger of Hearts catching their nearest rivals Ross County, with Hearts now 19 points adrift at the foot of the table.
BDO will also cite a perceived threat to the health and safety of any schoolboys who may be forced to play Premiership football if there is an injury crisis at Tynecastle.
Hearts' administrators are fearful that unless they become more competitive in the second half of the season home crowds could begin to dwindle - which would represent a serious financial problem in the club's efforts to survive until they exit administration, which may not be until April.
A number of players willing to turn out for the Jambos for a nominal fee are thought to be waiting in the wings.
"We will contact the SPFL on Monday and hopefully set something up for next week," said Jackson. "The circumstances have changed from the last time we spoke to the SPFL, so I need to make an effort and it is something that we need to try. We are not of any great threat, I suppose, to other clubs in terms of where we are in the league.
"And we are having to play these young boys so many times, through injury and all these kinds of things so, just for a bit of protection, I am going to ask the question again whether there is any assistance we can get."
There was, at least some good news for the beleaguered Hearts manager Gary Locke with the confirmation that his captain Danny Wilson could be fit to face high-flying Motherwell on Saturday.
The Scotland internationalist has been absent for the recent defeats against Hibernian and Partick Thistle due to a bout of tonsillitis, but the former Liverpool and Rangerts player has returned to light training this week.
Locke is hopeful that Wilson could be ready to lead out the Jambos at Tynecastle as they aim to halt a nine-match winless streak that has all but confirmed their relegation to the Championship.
He said: "It has been a blow not having Danny available but he has been back in the gym this week doing bits and pieces and has been feeling a bit better. He is a massive player for us. We'll see how he is during the week and we will give him every opportunity to make the match."
Locke club will also be heartened by the return from injury of Gary Oliver, the only natural striker in his ranks, who has been missing since suffering a sprained ankle against Inverness Caledonian Thistle last month.