WE are a Church with 501c3 status. We have been approached by an organization that was part of a University and as such used their 501c3 status for contributions etc. The organization lost the connection and have approached us (the Church) to house the organization in order to use our 501c3 status for the contributors. Apparently, their program is a federal program and will compensate the Church for the relationship. Is this allowed? Why or Why not?

Read about 'fiscal sponsors' here:
Fiscal Sponsors | National Council of Nonprofits

As for this particular group, I would want detailed proof regarding why they lost their status and proof showing exactly how the feds would compensate you. I wouldn't do anything without at least a minimal review by a lawyer - after this group is much more forthcoming, and they have to pay the fee. If they are just going around asking every church in town, you might be wasting effort. Make them do a lot more work.

I don't get it. Simply providing housing to an organization does not automatically extend your 501(c)(3) status to them. It would be helpful if you could give us an idea of what the organization is, as apparently it's either not a charitable, educational, or cultural organization, or has run afoud of the IRS's rules for a 501(c)(3). It may be possible for donors to make contributions to the church with the stipulation that it be used for the benefit of this organization - is that what they're asking you to do? If so, since this essentially involves the church making donations to the organization, the mission of the organization should be clearly aligned with the mission of the church. For example quite often churches will get involved in helping organizations such as day care centers, Boy Scout troops, summer camps, etc, which may be considered an extension of the church's mission regarding local community support. Clearly this is an item for discussion among the leadership of the church (the minister/priest/pastor as well as the elders, vestry, or whatever the "board of directors" is called in your church) to make sure it fits your mission.

I have known of groups within universities whose members all graduated and they failed to keep recruiting actual students, so they lost their status for that one simple reason. It might not be shady, but you haven't told us anything to go on.

To fall under your "status" they will have to merge with you, and your group take over control and management of the other group. Not merely "rent them space" will your board of directors have complete control of the new group ?

Also, if they are a university are they registered and licensed by your state (or some state where there corporation is set up). Remember, your church is not liable for their actions, once they become part of your group. So have you reviewed all of their practices.