The centre-back travelled to London at the weekend for a trial with the Premier League strugglers with the blessing of his football association, a week before his side begin their defence of the African title at the tournament in South Africa.
"The Zambia national team will always be there but opportunities like this do not come around often in a player's career so we were happy to let him go," said Football Association of Zambia president Kalusha Bwalya, himself a former professional in Belgium, the Netherlands and Mexico.
"We have agreed with Reading, his club TP Mazembe and ourselves that he takes up this chance. Reading were quite insistent on seeing him so it is a good sign," he told Reuters on Tuesday.
"Ultimately it is advantageous to our team to have players from clubs at the highest level."
Bwalya said it had been agreed that Sunzu would travel to South Africa by Friday.
"He will have enough time to have recovered from the journey by the time we play our first game."
Sunzu's long journey mirrors that of Burkina Faso winger Jonathan Pitroipa who has left his team's Nations Cup training camp to return to France to try to help Stade Rennes reach the French League Cup final.
Pitroipa is due to fly to South Africa, where Burkina Faso have been drawn in the same group as Zambia, on Thursday after Wednesday's semi-final against Montpellier.
Zambia, who have arrived at their base in Nelspruit, open the defence of their Nations Cup title against Ethiopia on Monday.
The 23-year-old Sunzu played every match of Zambia's successful run at the last finals in Equatorial Guinea and Gabon 12 months ago, converting the winning kick in the post-final penalty shootout that won his country their first Nations Cup title.