All three missed Sunday's 3-0 friendly defeat in Sweden but have resumed training at Wales' base in France.
Ledley's return is particularly significant, given the Crystal Palace player broke his leg on 7 May.
"If the situation stays the same, the three will be available this weekend," said Wales boss Chris Coleman.
Ledley resumed light training last week following the injury and his condition, along with the fitness of Robson-Kanu and Allen, has improved of late.
Robson-Kanu has had a leg injury while Allen has had a minor knee problem.
"All three players came through sessions yesterday, fully integrated with the squad, so it's positive news," Coleman continued.
"All three players are good, lacking game-time but physically they look very good."
Wales' match against Slovakia in Bordeaux will be their first in the finals stage of a major tournament since the 1958 World Cup.
Ledley, Allen and Robson-Kanu all played prominent roles in Wales' qualifying campaign but have seen their recent international involvement hampered by injury.
Wales' absence from major competitions means they were the fourth and bottom seeds when the Euro 2016 groups were drawn.
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Slovakia were third seeds after beating reigning European champions Spain on their way to qualifying.
Jan Kozak's side also claimed an impressive friendly victory away against World Cup holders Germany on 29 May.
"We know it's a tough test against a very tough Slovakian team, who will probably be favourites because they're a good team and had some great results," said Coleman.
"If you look at the four teams in the group, then we are the underdogs.
"We knew that before we got to the tournament. That doesn't mean we can't get enough points that we can't progress because we're capable of that - we know that.
"We know why we're here, but what's coming is something we've never experienced. It's a huge challenge for us, and one we deserve. This is what we've been screaming for."
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Midfielders Joe Ledley and Joe Allen and striker Hal Robson-Kanu are set to return from injury for Wales' Euro 2016 opener against Slovakia on Saturday.