WASHINGTON—A Russian-backed hacking effort that has rocked the presidential campaign may peak on Election Day, but is likely to continue next year and into 2018 as Moscow seeks to influence U.S. politics and key elections in Europe, Obama administration officials warn.

Russian meddling, in fact, may become more potent in Europe than in the U.S., White House officials and other experts say, because Moscow has long courted political figures there, and has forged ties with euroskeptic political parties strengthened by the recent...