IANAL, but this is plainly stated in the LGPL terms: see LGPL 2.1
§3. You may opt to apply the terms of the ordinary GNU General Public License instead of this License to a given copy of the Library. To do this, you must alter all the notices that refer to this License, so that they refer to the ordinary GNU General Public License, version 2, instead of to this License. (If a newer version than version 2 of the ordinary GNU General Public License has appeared, then you can specify that version instead if you wish.) Do not make any other change in these notices.
However, I do not see any particular reason to do this, except for derived works where you do not wish to continue to extending the additional LGPL privileges. The LGPL makes no specific requirement that it "be possible to link the program against a newer version of the LGPL library". The wording used is:
§6(b) Use a suitable shared library mechanism for linking with the Library. A suitable mechanism is one that (1) uses at run time a copy of the library already present on the user's computer system, rather than copying library functions into the executable, and (2) will operate properly with a modified version of the library, if the user installs one, as long as the modified version is interface-compatible with the version that the work was made with.
Nowhere is it required that a newer, not interface-compatible version, must work; that's hard enough as a technical goal, much less a legal requirement :-)
You're right that you cannot satisfy 6(b) without offering the user some means to install a modified version. However, You need satisfy only one of the section 6 subheadings, and part a, c, and d are the usual GPL "accompany with/written offer for/equivalent access to" source code options. These are further extended to permit that you may provide the non-derived, non-library parts of the work as object code, though this is certainly not FSF's preference.
So yes, you can use it under GPLv3 terms instead. But if you're willing to meet the full GPLv3 requirements, I don't really see what the problem is with meeting LGPLv2.1.