No - Richard was imprisoned by the Holy Roman Empire on the way home, and as you mentioned, Saladin died shortly afterwards on March 4, 1193.

Richard defeated Saladin at the Battle of Jaffa in August 1192, and they concluded a truce in which, among other things, Saladin kept Jerusalem but Christian pilgrims were free to visit the city. Most crusaders did go on pilgrimage throughout the rest of the year before returning home, but not Richard. He sailed for home on October 9, but his ship was wrecked and he was forced to make his way through the territory of the Holy Roman Empire. Unfortunately for Richard, he had managed to offend the Duke of Austria, who had also been on crusade with him, by refusing to let him fly his banners when they captured Acre. Earlier in April 1192, the crusader Conrad of Montferrat was elected as the new King of Jerusalem, but he was killed by the Assassins almost immediately. Conrad was a cousin of the Duke of Austria and the Emperor, so they blamed Richard for the murder. The duke took Richard prisoner and handed him over to the emperor.

Richard was kept as a hostage until February 1194, when he was released for an enormous ransom of 150,000 marks, which had to be raised through a special tax in England. Meanwhile, Saladin spent time with his family in Damascus and Jerusalem. He made plans to return to Egypt, and was even thinking about making the pilgrimage to Mecca, but he never made it. He fell ill at the end of February 1193 and died in early March. During this time, Saladin did meet with the crusader Prince of Antioch, and some ambassadors from the crusader pilgrims who were still present, but he never communicated with Richard again.

In fact the two never actually communicated in person even while Richard was still in the east. They only communicated through ambassadors who travelled back and forth between their two camps. Their immense respect for each other may be a bit an exaggeration too…they certainly had difficulties communicating sometimes. At one point after the siege of Acre, Richard executed several thousand prisoners instead of ransoming them, and in retaliation Saladin killed numerous crusader prisoners. Their negotiations after the Battle of Jaffa were not exactly totally friendly; Richard wanted Jerusalem back, of course, but Saladin was obviously never going to hand it back. Their communication was sometimes angry and sometimes even sarcastic. The idea that they were two perfectly chivalrous knights who respected each other is more of a modern invention.

There is one Muslim leader that Richard may have contacted while he was in prison. He presented the emperor with a letter, which was supposedly written by the “Old Man of the Mountain”, the leader of the Assassins. The Old Man claimed responsibility for Conrad’s death and absolved Richard of any knowledge. This may have helped convince the emperor that Richard was innocent (although he still kept Richard in prison). But the letter was presumably a forgery so Richard never actually communicated with the Old Man either.

Sources:

John Gillingham, Richard I (Yale University Press, 1999)

M.C. Lyons and D.E.P. Jackson, Saladin: The Politics of the Holy War (Cambridge University Press, 1984)

Jonathan Phillips, The Life and Legend of the Sultan Saladin (Penguin, 2019)