The earnestly built facade of cooperation that City Council Chairman David A. Clarke has fashioned around council deliberations since his election last year is begining to show some signs of wear.

Clarke's effort to juggle the sharply different personalities of the other 12 members--and to dampen his own temperament--has resulted in several important victories, but at the same time it has produced strains that could affect the council's ability to deal with major issues.

Clarke says such tensions among council members are natural and the true test of his leadership--as he stressed in his campaign last year--is whether the council ever lapses into the petty bickering and sniping that characterized many sessions under former chairman Arrington Dixon.

Although that has not come close to happening, some members recently have engaged in rhetorical attacks on fellow council members that were not evident earlier this year and suggest that all is not as smooth as Clarke would like.

One of the most potentially serious problems for Clarke is a flap that has developed between him and John A. Wilson, the veteran Ward 2 Democrat, who has played a de facto vice chairman's role early in Clarke's first term as chairman.

"He has lost a good friend," Wilson has noted privately, saying Clarke purposely misled him recently on the existence of a sweeping legislative plan, which Wilson opposes, that would consolidate elections and put off this fall's school board balloting.

Wilson, who as head of the finance and revenue committee helped Clarke forge a cooperative partnership to pass the fiscal 1984 budget with relative ease, contends Clarke attempted "an end run" by saying the plan didn't exist in written form while at the same time trying to round up seven votes.

While Wilson and Clarke don't expect to agree on every issue, Wilson's relationship with Clarke is crucial because Wilson can skillfully influence the votes of several other members or make Clarke's life miserable.

Clarke, who says the election flap was a misunderstanding, was concerned enough about it at one point to use his auto's flashing red light to rush from his Mount Pleasant home to the District Building to soothe Wilson's feelings. But Clarke said last week he would not discuss the incident "in the press."

Clarke is said to believe the dispute with Wilson, one of the most moody and mercurial council members, should fade as they approach sticky issues, such as the repeal of no-fault auto insurance and condominium conversion--areas in which they agree.

Oddly, one of the reasons for Clarke's trouble is that some members believe he and Wilson together have been too successful.

"John is starting to believe his press," one member has said more than once to reporters. Another recently talked for more than an hour about an alleged conspiracy by the media to "build up" Clarke and Wilson at the expense of other members.

Such criticisms also reflect the undercurrent of racial sensitivity to the fact that Clarke is a white chairman of a majority-black council in a majority black city, while Wilson, who is black, runs the risk of appearing too cooperative.

In that context, some observers believe Wilson could be using the dispute with Clarke to put a little distance between the two.

Clarke, who acknowledges the problem of "getting out front" of the members--for which Dixon was criticized--said he has not sought favorable publicity. "You haven't seen press releases coming out of this office," he said.

Clarke also has made a point to lavishly praise council members at public events and to defend them, such as when six members took a free trip to Germany two weeks ago.

Despite his efforts, however, Clarke is often warily viewed by some members who have variously criticized his intense, sometimes arrogant manner in stating his views.

As one longtime staff member said, "Black, purple or white or right, David is just hard to relate to."