Sen. Herman E. Talmadge (D-Ga.), whose finances are still under investigation by the Senate Ethics Committee, has repaid the Senate $37,125.90 in improper expense claims his office said yesterday.

Almost immediately, however, questions were raised over who had the authority to determine the interest due on the funds - an amount that could easily run to several thousand dollars. Although Talmadge offered to pay interest, it appears there may be no rule covering the matter.

A personal check from Talmadge was hand-delivered to Senate secretary J. Stanley Kimmitt on Friday evening, along with a letter explaining that the payment was for expenses collected "in excess of reimbursable amounts" from 1972 to 1977.

The money includes claims for more than $24,000 in expenses that were never incurred and more than $11,000 for expenses not considered reimbursable under Senate rules. Talmadge has blamed the claims on "errors in judgment" by staff members who handled his finances.

The Ethics Committee decided at a closed meeting on Thursday to accept Talmadge's payment, but a committee spokesman stressed yesterday that the repayment would not affect the committee's own investigation into Talmadge's affairs.

"The committee had not requested repayment, but decided it was his prerogative if he wanted to pay," the spokesman said.

The figure of $37,125.90 was determined by auditors hired by Talmadge after allegations about the improper expense claims were raised in the press.

Investigators for the Ethics Committee have also examined the expense claims. Their findings will be relased when the committee concludes its report on Talmadge.

Talmadge noted in his letter to Kimmitt that the refund does not include interest, but added, "if you will advise me as to the amount of interest due the Senate because of these excess payments. I will promtply forward this additional amount to the Senate."

Kimmitt said yesterday, however, that he is not the proper person to determine the interest due, and, in his opinion, it is enough to repay the principal.

Senate Majority Leader Robert C. Byrd (D.W.Va.) disagreed, however, saying the Ethics Committee should determine the amount of interest due.

"That's a matter for the Ethics Committee to decide," Byrd said."If he has volunteered (to pay the interest), that should end it."

Ethics Committee staff members, uncertain of th committee's authority concerning interest, are researching federal statutes on the question.

Talmadge, who has said he will run for reelection in 1980 with a clear conscience, also noted in the letter that the repayment does not include about $13,000 claimed by a former aide for expanses never incurred.

The former aide, Daniel Minchew, now a member of the International Trade Commission, has said he filed for the expenses at Talmadge's direction and then gave the money to the senator and his family.

Talmadge disputes Minchew's statement, and said in his letter to Kimmitt that Minchew's claims for reimbursement "were procured without my knowledge or consent and were not deposited in my special account in a Georgia bank.

Talmadge also notes in the letter the matter has been turned over to the Justice Department and the Ethics Committee for investigation.

Carl Eardley, special counsel for the Talmadge investigation, has said his work might be concluded by the end of next week.

The full committee would then review the work of its staff and conclude whether further actions with respect to Talmadge are neccessary.

The committee will meet tomorrow afternoon in closed session, but then no further meetings are scheduled until after Labor Day.

[The Ethics Committee asked the Justice Department yesterday to grant limited immunity for Michew in order to gain his testimony in the investigation, United Press International reported.]

[If the department agrees, it would free Minchew from prosecution based on any testimony he might give to the committee in its investigation. Minchew's lawyer sought the immunity. The department is to respond to the panel's request for it on Sept, a committee spokesman said.]