Clarke, who took over at The Hawthorns in June 2012, is out of the contract at the end of the season.
The 50-year-old replaced Roy Hodgson, after he had been appointed England manager, at the Black Country club in what was his first managerial role having previously been assistant manager at Chelsea, West Ham and Liverpool.
The Baggies are currently 10th in the table, with Clarke having guided them to eighth last season.
In a statement on the club's official website, sporting and technical director Richard Garlick said: "Steve's contract situation was discussed at last month's board meeting.
"We decided that we would open talks with Steve in the new year which, historically, we feel is the appropriate time to address such matters."
Thursday evening's announcement follows the news the club are also in talks with England Under-21 striker Saido Berahino about a new deal.
The 20-year-old has been one of the breakthrough stars of the season and the Baggies are keen to tie him to a new long-term contract.
His current contract runs out at the end of the season, although West Brom do have the option of extending it for another year.
Press Association Sport understands that reports stating Berahino is about to sign a new deal are premature, but talks are progressing well.
Berahino came to England aged 10 after escaping war-torn Burundi with his mother and siblings and joined West Brom a year later.
He signed his first professional deal at The Hawthorns in 2011 and spent time on loan with Northampton, Brentford and Peterborough.
Berahino made his first-team debut for the Baggies as a late substitute in a Capital One Cup tie against Yeovil in August 2012, and it was in the same competition earlier this season that he really made his mark.
The striker netted a first-half hat-trick against Newport County and then scored the equaliser against Arsenal in the same competition in a game West Brom eventually lost on penalties.
Berahino's stock rose further three days later when he scored his first Premier League goal at Old Trafford to help his side to a famous win over Manchester United.
Earlier this month he scored again against Crystal Palace, and the 20-year-old's exploits have reportedly alerted some of the Premier League's big guns.
Berahino first played for England in 2009 at under-16 level and has moved up through the age groups.
He scored on his debut for the under-21s against Moldova in September and has since taken his tally to six goals from four games.
Berahino's most recent strikes came against Finland last week, but he missed the 9-0 win over San Marino after being booked for revealing a T-shirt with a tribute to his late father.
Under-21 boss Gareth Southgate heaped praise on the young striker, saying: "He is a good player, a good finisher and a pleasure to work with.
"He has got a desire to improve all the time and he is a terrific goal threat for us."