There were two notable attempts by the papacy to broker a peace deal. In January of 1915, Benedict XV sent a papal diplomat as an envoy to Austrian emperor Franz Joseph. That diplomat was Eugenio Pacelli (who in 1939 would be elected pope and would take the name Pius XII). The goal was to try to keep Italy out of the war by having Austria agree to Italian territorial demands. This initiative was a failure.

In 1917, Pacelli was again selected to try to get Kaiser Wilhelm of Germany to agree to Benedict XV's 7-point peace plan. As Robert Ventresca says,

[i]n the end, the discussions of the summer of 1917 went nowhere. Benedict XV's peace plan was effectively dead in the water. Despite some promising starts made by Pacelli in his first meeting with the German chancellor, the German military high command was in no mood for the concessions Bethmann-Hollweg had seemed ready to accept earlier that summer. (Soldier of Christ: The Life of Pope Pius XII*, pg 48)

So, early papal efforts to call for peace and keep the war from expanding were unsuccessful, as was the 1917 7-point proposal by Benedict XV.

Woodrow Wilson made a number of attempts to mediate some sort of peace agreement between the Entente and the Central Powers. This ran into obstruction and denial of varying levels from both sides. Wilson (never a fan of Britain or placing much emphasis on US-UK relations) abandoned any hope of constructing peace on a basis of close relations with the UK and apparently even toyed with an alliance with Germany simply to fight the UK.

Lloyd George also attempted to make peace a number of times over the course of his wartime coalition government, but never with Germany. Instead, given his preference for attempting to avoid bloodshed on the western front as much as possible to preserve the state of the British arm for the final, victorious campaign, he hoped constantly to sway Bulgaria, or the Ottoman Empire, or Austria-Hungary to make peace. In this he was obviously unsuccessful. However, at times the British (largely the Foreign Office, of whom Ll.G. tended to act independently as he loathed diplomats) still considered peace with Germany, sometimes on the basis of accepting (some) German territorial claims in the east (such as Latvia and Lithuania but not Estonia or Finland) in return for ceding Alsace-Lorraine back to the French.

Source: British War Aims and Peace Diplomacy, 1914-1918, Victor H. Rothwell.

Ultimately, the nature of coalition warfare means that coalitions tend to fail only when their strongest member capitulates. In WW1, that was Germany for the Central Powers and Germany was only willing to capitulate (rather than impose terms) after the hundred days and the collapse of the German army in autumn 1918.

It may not be quite what you mean, but there were attempts to avoid the first world war by diplomats on both sides of the conflict. The time between the assassination of Franz Ferdinand and the cascading of war declarations in August is known as the July Crisis. During the July Crisis, efforts were made by Edward Grey, the British foreign secretary, to mediate the Austro-Serbian dispute. He proposed that Austria-Hungary occupy Belgrade as a temporary solution without declarations of war or ultimatums. He also was hoping for international cooperation in this.

Source: Great Power Diplomacy, Norman Rich