WASHINGTON — The Brady Campaign to Prevent Gun Violence, one of the nation’s leading gun control groups, said Friday that it wanted the White House to focus its attention on expanded background checks for gun buyers as part of a broad push to reduce gun violence in the wake of the school attack in Connecticut last month.

The group made the recommendations to Vice President Joe Biden. The New York Times obtained a copy of the document, which stresses that ‘‘closing the massive hole in the background check’’ system is the group’s top policy priority.