The Welsh side also came close through Marc Guehi, who blazed over the bar from six yards out.
But the Berkshire side held firm for a point and a clean sheet, but they still had their chances to win the game, most notably when Tom McIntyre forced a save from Freddie Woodman with a powerful header.
Story of the match
Swansea came into this game with the best defensive record in the Championship, losing once in their last six games. They made one change from the starting eleven that beat Queens Park Rangers on Boxing Day, with experienced midfielder Korey Smith coming for Yan Dhanda.
Reading managed to extend their Boxing Day unbeaten run with all three points over Luton Town, and also only made one change for this game, with Michael Olise coming in for Sone Aluko. Omar Richards also came back to the Royals bench, after missing the last three games with a hamstring injury.
In a relatively quiet opening 15 minutes, the first piece of the action was when a cross was put in for Swansea star Ayew, who tried an audacious bicycle kick, only for it to end up going over the bar.
Reading has their first chance when Olise whipped in a beautiful ball into the box which McIntyre met on the volley, only to force a fantastic save from Woodman, as he was desperate to claim his 13th clean sheet of the season.
Ayew tried again to test goalkeeper Rafael, shooting from 30 yards out, only forcing the Brazilian to make a comfortable save to keep the Swans at bay.
Swansea kept pushing for a goal, and it was the third time unlucky for Ayew, as Connor Roberts played a high cross in for an open Ayew, who headed and forced a wonderful save from Rafael, as the Swans kept pushing for the lead.
Swansea then came close again, as a corner was knocked down to the edge of the six-yard box for an open Guehi, but the Chelsea loanee blazed the ball over the bar.
The second half started with Swansea carrying on their momentum but Reading posed more of a threat, but Joel Laitbeaudiere, having just come on for Ryan Bennett who came off with a muscular injury, leaped highest from a cross from the left-hand side and managed to head down to the bottom right corner, before Rafael was there yet again to keep the game goalless.
Reading them made a tactical change in bringing on John Swift for Sam Baldock, meaning they had no established striker on the pitch.
Rafael was yet again forced to make another save, as Roberts put another cross in for Ayew who headed towards goal and Rafael used his excellent reflexes to deny Swansea what looked to be their first goal of the night.
Reading started to press on as they neared the full-time whistle, with McIntyre coming inches away from scoring with a wonderful first-time header which was tipped over by Woodman.
The game ended goalless, which reading will take better than Swansea.
Man of the match: Rafael (Reading)
It was a close call between him and Andre Ayew, who tested the Brazilian a number of times, but it was Rafael who got the better of him, so for that reason he is my man of the match.
Rafael kept Reading in it from start to finish, with some fantastic saves to keep Swansea and more notably Ayew at bay, and deserves the plaudits for their first clean sheet in four games.