The 28-year-old Warriors centre, who came to Sixways directly from Rugby School in 2006, has largely been limited to European Challenge Cup appearances this season.
Grove, who played a big part in getting Warriors promoted in May, has failed to make Dean Ryan's Premiership team.
He will join utility back younger brother Oli, 25, at the Exiles.
The three-times capped Scotland centre, who made his international debut in 2009. has run in 29 tries in 151 appearances for Warriors,
Grove spent six months on loan to Edinburgh for the first half of the 2010-11 season, following Worcester's relegation to the Championship.
He was then part of the Warriors team which went down again in 2014, but stayed at the club this time, playing a key role in their return to the Premiership via the play-offs, which included two semi-final victories over two legs against Scottish.
"I spoke to Dean Ryan about getting some more time out on the pitch and I'm familiar with the club after coming down many times to watch Oli play," said Gove.
"It's a great opportunity to get some game time at Scottish and hopefully contribute in the coming weeks."
Worcester Warriors have loaned former Scotland international Alex Grove to Championship side London Scottish.