WASHINGTON (CNN) Republicans have begun to sour on Robert Mueller's handling of the investigation into Russian interference in the 2016 presidential election, according to a new CNN poll conducted by SSRS , and they are increasingly likely to say the President should not testify under oath for the special counsel.

Overall, 44% approve of the way Mueller is handling the investigation and 38% disapprove. That's a slight negative shift overall since March, when 48% approved of Mueller's work. Just about all of that change has come from Republicans, who now give Mueller a 17% approval rating, down from 29% in March. Among Democrats and independents, approval ratings for Mueller have not changed significantly.

And Republicans have rapidly turned against the idea of the President testifying under oath before Mueller. Although 70% overall say Trump should testify for Mueller's investigation if asked, that stands at just 39% among Republicans, down from 54% saying the same in March. Among independents and Democrats, support for Trump testifying has held roughly steady compared with previous CNN polling on the topic. About 6 in 10 (62%) overall say Mueller ought to subpoena the President if he won't agree to speak with investigators.

In the CNN poll, the President's approval rating for handling the investigation remains deeply negative, with just 31% approving of the way he's handling it and 55% disapproving. Those figures haven't moved significantly since CNN and SSRS first began polling on the question last August.

But slightly more now believe the things the President has said publicly about the investigation are true than said the same in December. Among all adults, 39% say Trump's statements on the investigation have been mostly or completely true, up from 35% in December. While three-quarters of his partisans (75%) say he's mostly said true things about Russia publicly, most Democrats (84%) and independents (52%) believe his statements have been largely false.

More Americans say they believe the things former FBI Director James Comey has said publicly about the investigation, according to the poll conducted not long after the release of Comey's much-anticipated book addressing his relationship with Trump. About half of Americans (49%) say Comey has mostly said true things publicly about the investigation vs. 37% who say his statements have been mostly false. Party divides are steep on perceptions of Comey, with 70% of Democrats saying they think Comey has mostly said true things about the investigation while just 21% of Republicans agree.

A vast majority of Americans -- 84%, including majorities across party lines (93% of Democrats, 86% of independents, 72% of Republicans) -- say they think the investigators should produce a full, public report on their findings regardless of the outcome of the investigation. Just 13% say those findings do not need to be made public. And about 6 in 10 (58%) continue to say the investigation is a serious matter that should be fully investigated, while 38% say that it's more an effort to discredit Trump's presidency.

Looking to Russia itself, an increasing share say the country is a "very serious threat" to the US; 38% say so now, up from 31% when we last asked in October. That stems from increases among Republicans (from 19% in October to 28% now) and independents (from 26% in October to 39% now); Democrats' views on the threat from Russia have held relatively steady.

Russian President Vladimir Putin remains deeply disliked in the US, with 77% expressing unfavorable opinions, up from 71% saying so just before Trump's inauguration in January 2017. That increase comes largely from a shift among Republicans, who had softened in their views of Putin around the time of Trump's inauguration. In the new poll, 75% of Republicans say they have unfavorable takes on Putin, up from 56% in January 2017. Negative views among independents have also risen, up 7 points to 76%, while among Democrats, unfavorable views now (85%) are about the same as in 2017 (86%).

The CNN Poll was conducted by SSRS May 2-5 among a random national sample of 1,015 adults reached on landlines or cellphones by a live interviewer. Results for the full sample have a margin of sampling error of plus or minus 3.6 percentage points; it is larger for subgroups.