The issue is why you are living in Michigan while working for a New York company. 

If you are in Michigan "for the convenience of the employer" - that is, because your employer wanted you there rather than in New York - then your wages would be taxable in Michigan.

However, for most employees in your situation who are living in another state for their own convenience and not at the employer's requirement, then you are required to report your wages as New York source income, taxable in New York on your NY nonresident return.

Do your NY return first, then do your Michigan resident return. In this way, your NY tax will find its way to line 18 on your Michigan MI-1040, to reduce your Michigan tax owed.

Thank you! So far the employer is saying that deducting taxes for New York isn't necessary, only Michigan taxes need to be deducted. I don't want to get a surprise at the end of the year and have to pay New York thousands of dollars and possibly a penalty of some sort! Is there a link or something that I can forward to them to reference? 

Don't try to overanalyze this - if your employer is reporting your W-2 earnings as "MI" sourced (ie. Boxes 15-17 only refer to MI, not NY), then file your MI return and don't worry about NY.

It is a long-standing principle that an employee's wages are taxable in their state of residence and in the state where they perform their work. Since you performed the work in MI and are a resident of MI, then your wages are definitely taxable by MI.

A minority of states, however, utilize convenience of the employer (COE) rules to determine how nonresident remote employees should be taxed on their income.  These states currently include Connecticut, New York, Pennsylvania, Arkansas, Delaware and Nebraska.

As @BillM223 stated, "If you are in Michigan "for the convenience of the employer" - that is, because your employer wanted you there rather than in New York - then your wages would be taxable in Michigan."  It is reasonable to presume your employer is aware of NY rules and has concluded you are working in MI for their convenience.

You can ask your employer to confirm this - send them this article for a detailed discussion and the rules on What is New York's "Convenience of the Employer" Rule All About?

Follow-up question on this...

I've been in Michigan, working for a NY company at my convenience for several years and every year I have issues with my NY return. TurboTax asks you to quantify how many days you worked in NY vs. MI, which I do (hotel receipts make this very easy). It's normally 10% NY and 90% MI, so TurboTax projects a large return from NY. However, EVERY year, NY audits me and adjusts the return to be lower and I can't figure out why. 

It's frustrating because if I knew in advance exactly how much NY would adjust my return for, I would just submit the original as if I spent that many days in the state (because the result is me getting taxed in MI as if I worked there 90% of the year AND taxes in NY as if I were there, say, 40% of the time instead of the 10% I submitted). 

@mackenzze --

Since your employer is based in New York, and since you perform your work both within and without the State of New York, your work income is subject to New York's "convenience of the employer" doctrine.  Here is NY's rule:

"However, any allowance claimed for days worked outside New York State must be based upon the performance of services which of necessity, as distinguished from convenience, obligate the employee to out-of-state duties in the service of his employer."

https://www.tax.ny.gov/pdf/memos/income/m06_5i.pdf

By your own statement, the work you do in Michigan is done there for your own convenience.  The convenience rule means that, for tax purposes, New York will treat the days you worked in Michigan as if they were days worked in New York.  That means, even though you physically work in New York only 10% of the time, ALL your work income is taxable by New York.  It is also taxable by Michigan, since Michigan residents are taxed by Michigan on their entire income, regardless of its source.

You will be able to claim an "other state credit" on your Michigan tax return for the taxes paid to New York on the dollars taxed by both states, so in effect you won't be double-taxed.

With TurboTax, be sure to complete your non-resident NY tax return before you do your home state MI return, so that the program can correctly calculate and apply the credit.   Finally, be sure your employer withholds NY taxes from your pay.