It doesn't typically take a very long time for the IRS to process Form 843 but an exact time frame isn't known.
An employee can't claim a social security overpayment on their tax return because there is nowhere to claim it. Therefore, no software will offer that option.
Typically, an employer is happy to adjust the overpayment because it would mean that they also overpaid and they would be due their own refund. However, you certainly can only control your side of things, and you cannot make your employer apply for their own refund.
The only ways that an individual can reclaim an overpayment of social security are from the employer or by filing Form 843 separately from the tax return, when the employer will not adjust the overpayment.
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