The Tynecastle Park side were decimated with injuries during their 2-2 draw with Dundee United at Tannadice on Saturday in the Scottish Premiership.
Alex Cochrane, Gary Mackay-Steven, John Souttar, Cammy Devlin and Nathaniel Atkinson were all replaced due to concussion, broken hand, ankle knock, hamstring complaint and illness respectively.
On top of that, Josh Ginnelly received treatment on the pitch at full-time, while Beni Baningime and Peter Haring also felt discomfort.
Players will be given until the last minute to prove their fitness but the likelihood is they will be without Cochrane, Mackay-Steven, Souttar and Devlin, as well as Michael Smith.
Neilson believes running with a smaller squad will prove to be a benefit as the club look to book their place in their fifth cup semi-final in the last four seasons.
“We’ve had a really good fitness percentage throughout the season then we have a wee period where we have got a number of injuries," he said. “Some are a day, some will be two, some will be three days, some will be three weeks. We just need to deal with it.
"I’m not going to go into the details of all the players because we’ll be here all night!
“There are still a few that are borderline. Some will come in on Saturday morning and we’ll test them, we’re hoping those extra days recovery will be fine.
“The benefits of [a smaller squad] is that the 11 that start on Saturday will all be top three, top four players in Scotland. We don’t have a 24-man squad where we have ten really good players and ten ones that are miles off it. I don’t have any worries about it.
"We’ll have a strong 11, how many we have on the bench will be a different story. It might be three, it might be six.”
The Scottish Cup holds positive memories for Neilson, having won the tournament as a player with Hearts in 2006. He would make history by leading Hearts to success as manager, with three games to navigate.
He feels the prestige of winning it matters more than the European benefits with the Cup winners gaining entry into the Europa League play-off and guaranteed group stage football.
“It doesn't influence it at all because it is the Scottish Cup,” he said. “Rewards we get off the back of that are great, but we want to win it.
"I'm not interested in how it gets you into Europe or what it does with Europe. Most managers and players will be the same, it's about winning the Scottish Cup.
“There's only four [Hearts] teams who have won the Scottish Cup in the last 65 years. A lot of them have qualified for Europe.
"I spoke to the players about it. The season is about qualifying for Europe, brilliant. But you want to win something. You want to win the Scottish Cup.
"For me it's not got any influence on European football. It's about winning.”