Williams pulled out of the game against England with a shoulder problem, while Anscombe limped off in the first-half at Twickenham with a right knee problem.
“I spoke to the physios and Gareth has got to get a scan. If it is just a strain of the ligament then he will be out for a few weeks, but we will know more about the details after he has had the scan,” said Gatland.
“Tomos Williams is being scanned on Monday, too. Adam Beard has a bit of a wrist strain, but he’s been x-rayed and that’s clear, and Justin Tipuric had a slight roll on the ankle, which I don’t think is an issue.”
England’s 33-19 triumph ended Wales’ record 14 match winning streak and also stopped them from taking over as the No 1 ranked team in the world. But Gatland is more concerned about next weekend’s performance against Eddie Jones’ English side than the record books.
“It’s a nice accolade in terms of being number one in the world, but the biggest prize is in a couple of months’ time in Japan. Whoever wins the World Cup will be No 1,” said Gatland.
“It’s disappointing in terms of the result, but it’s not something we will dwell on too much. It’s about what we take out of today.
“There are things to work on and to address. There were also some positives from today as well and we need to make sure that next week’s performance is an improvement on today.
“We need to look at ourselves in terms of a few penalties we gave away around the fringes. We need to address that, as it gave them some soft three points.
“When you start the season with a Test, you are straight into it. We are fortunate to have another one next week and our review and preview processes will be exactly the same.”
“We got into the speed of the game after a while and were able to impose ourselves. In terms of being rusty, there were just too many turnovers, which meant we didn’t keep possession for long-enough periods and we didn’t get the momentum we needed.
“One of the stats from our analysts were that every time we made a break the result of the next couple of phases were that we either scored or were turned over. We need to be a little better with that, and I’m sure we will be.
“They started well, but the disappointing thing for me was the try at half-time with the overthrow. We had got back into the game at 24-19, started to create a lot of momentum and I thought we were going to win the game from there.”
Welsh skipper Alun Wyn Jones, who overtook Gethin Jenkins as Wales’ most capped player as he played his 135th international for Wales and the British & Irish Lions to move on his own into fifth place in the world list, admitted his side were rusty.
“We’ve had these questions in the Six Nations and the autumn about starting poorly and you could say we started poorly in this year’s Six Nations. We did pretty well in the end, though,” said Jones.
“We prepared well, but there are things to address which we would like to think are all fixable. The buzz word is momentum – we felt we were pressurising them a lot at lineout time, but we didn’t convert that pressure.
“Likewise at the scrum, when they had a good foundation we were chasing to stop their momentum. There are things we can clearly work on for next week, but we cannot rely on saying we were rusty because it feels like an excuse saying that.
“When you start the season with a Test, you are straight into it. We are fortunate to have another one next week and our review and preview processes will be exactly the same.”
