There are certain situations under which a statute of limitations will toll or be reset. Under the facts that you shared in your question, none of those facts seem to be present. In fact, not only is your debt likely stale and unenforceable in court, the debt collector seems to be close to committing violations of the federal and state of texas unfair debt collection acts by making misrepresentations about his rights and your obligations. Even if the debt were valid and enforceable it is highly unlikely that they have any right to make you take out a loan or have any basis to request a bank routing number. If the attempts continue or becomes harassing, you may have a cause of action and may consider contacting an attorney.
Threats to sue after the statute of limitations has run are illegal. What property? Threats to put a lien on a homestead are illegal. What collection agency? Someone in state or out of state? If it was a law firm in state, they may actually intend to sue you, but you will also get a letter from the law firm stating as much. If it is an out of state collector, they have no authority to file a suit and the collector is lying to you. Again, a violation of the law. Never give a debt collector bank information.
Nothing you described would restart the statute of limitations in Texas.
Get the collectors name and address and send them a letter telling them to stop all attempts to collect the debt. Also, get your credit report and see what, if anything, they are reporting on your credit.
I am licensed only in Texas. Offering information of a general nature in response to a question is not intended to be legal advice in your state.
You can restart the statute of limitations if you, in WRITING, agree to repay the debt or acknowledge that you owe the debt. It would be like you created a new debt. In your situation your best legally permissible course of action is to NOT make any payment arrangements and to write a letter to the bill collector asking them to stop contacting you about the debt. Their only option would be to sue you (or sell the debt to someone else). Of course, you could go ahead and pay the debt or settle for a portion of it if you feel morally compelled to do so.